Multiproduct Firms and Price-Setting:
Theory and Evidence from U.S. Producer Prices∗
Saroj Bhattarai and Raphael Schoenle†
Pennsylvania State University and Brandeis University
October 2013
(First version: Aug 2009. This version: October 2013)
Abstract
In this paper, we establish three new facts about price-setting by multi-product firms and
contribute a model that can explain our findings. Our findings have important implications for
real effects of nominal shocks and provide guidance for how to model pricing decisions of firms.
On the empirical side, using micro-data on U.S. producer prices, we first show that firms selling
more goods adjust their prices more frequently but on average by smaller amounts. Moreover,
the higher the number of goods, the lower is the fraction of positive price changes and the more
dispersed the distribution of price changes. Second, we document substantial synchronization of
price changes within firms across goods and show that synchronization plays a dominant role in
explaining pricing dynamics. Third, we find that within-firm synchronization of price changes
increases as the number of goods increases. On the theoretical side, we present a state-dependent
pricing model where multi-product firms face both aggregate and idiosyncratic shocks. When we
allow for trend inflation and a menu-cost technology that is firm-specific and features economies
of scope, the model matches the empirical findings.
JEL Classification: E30; E31; L11.
Keywords: Multi-product firms; Number of Goods; State-dependent pricing; U.S. Producer
prices.
∗
We thank, without implicating, Fernando Alvarez, Jose Azar, Alan Blinder, Thomas Chaney, Gauti Eggertsson,
Penny Goldberg, Oleg Itskhoki, Nobu Kiyotaki, Kalina Manova, Marc Melitz, Virgiliu Midrigan, Emi Nakamura,
Woong Yong Park, Sam Schulhofer-Wohl, Kevin Sheedy, Chris Sims, Jon Steinsson, Mu-Jeung Yang and workshop and conference participants at the Board of Governors of the Federal Reserve System, the CESifo Conference
on Macroeconomics and Survey Data, the Midwest Macro Meetings, Recent Developments in Macroeconomics at
Zentrum für Europäische Wirtschaftsforschung (ZEW) and Mannheim University, the New York Fed, Princeton University, the Swiss National Bank, the Royal Economic Society Conference, and the XXXV Simposio de la Asociación
Española de Economı́a for helpful suggestions and comments. This research was conducted with restricted access to
the Bureau of Labor Statistics (BLS) data. The views expressed here are those of the authors and do not necessarily
reflect the views of the BLS. We thank project coordinators, Ryan Ogden, and especially Kristen Reed, for substantial
help and effort, as well as Greg Kelly and Rosi Ulicz for their help. We gratefully acknowledge financial support from
the Center for Economic Policy Studies at Princeton University.
†
Contact: Saroj Bhattarai, The Pennsylvania State University, 615 Kern Building, University Park, PA 168023306. Phone: +1-814-863-3794, email: [email protected]. Raphael Schoenle, Mail Stop 021, Brandeis University, P.O.
Box 9110, 415 South Street, Waltham, MA 02454. Phone: +1-617-680-0114, email: [email protected].
1
Introduction
Using micro-data underlying the U.S. Producer Price Index (PPI), we provide new evidence on
several key statistics of price changes systematically varying with the number of goods produced
by multi-product firms. Our main empirical findings are that as firms produce more goods: the
frequency of price change is higher while the size of price changes is lower; the fraction of price
changes that are increases is lower; the fraction of small price changes and the dispersion of price
changes is higher; and the degree of synchronization of price changes within firms increases. As a
byproduct, we document a substantial degree of synchronization of price changes across products
within a firm, relative to synchronization with other firms in the same sector. We then present a
multi-product menu cost model that can match our key empirical facts.
Our findings are important because they can help us distinguish between competing models of
price adjustment, and in particular, gauge the extent of the so-called “selection effect” in menu cost
models. Making a distinction between models of price adjustment is of first-order importance in
macroeconomics: even small changes in assumptions about price setting can imply vastly different
real effects from nominal shocks. Our analysis of how the number of goods in a firm relates to pricing
decisions is moreover of independent interest: since approximately 98.55% of all prices in the PPI
are set by firms with more than one good, this fact contrasts with the standard macro-economic
assumption of price-setting by single-product firms.
To arrive at our results, our analysis exploits a particular feature of the micro data from the
Bureau of Labor Statistics (BLS) that underlie the calculation of the U.S. PPI. In these data, there
is substantial variation in the number of goods per firm, with a median number (std. deviation) of
4 (2.55) goods. This feature enables us to compute price change statistics according to the number
of goods per firm.
While doing so, we find that pricing behavior is systematically related to the number of goods
per firm. First, this holds true especially for the frequency and size distribution of price changes, two
key statistics that can discipline the real effect of monetary policy shocks. As is well-understood,
higher frequency of price changes leads to lower real effects on monetary shocks in models with
nominal rigidities. We find that the frequency of price adjustment increases monotonically and
2
substantially as the number of goods increases. For example, firms that have between 1 to 3 goods
on average during their time in the data have a median monthly frequency of price change of 15%.
Firms that have more than 7 goods have a median monthly frequency of 23%. At the same time,
the average magnitude of price changes, conditional on adjustment, monotonically decreases with
the number of goods. This result holds for both upwards and downwards price changes. The
magnitude of price changes ranges from 8.5% for firms with 1 to 3 goods to 6.5% for firms with
more than 7 goods.
Two other statistics – the fraction of small price changes and their kurtosis – are key moments
related to the importance of selection effects in menu cost models. We find that small price changes
are highly prevalent in the data, and become even more prevalent when the number of goods
increases. The fraction of small price changes is 38% for firms with 1 to 3 goods and increases to
55% for firms with more than 7 goods. We also find that there is substantial dispersion in the size
of price changes that increases with the number of goods. For example, the coefficient of variation
of absolute price changes and the kurtosis of price changes increase as the number of goods per
firm increases. The mean kurtosis for firms with 1 to 3 goods is 5, and 17 for firms with more than
7 goods.
In terms of macroeconomic implications, such evidence directly relates to the strength of the
“selection effect” in menu cost models: Firms that have prices that are far from their optimal prices
are much more likely to adjust prices given adjustment frictions, especially when a shock pushes
prices even further from their optimal levels. The stronger this effect, the more prices absorb shocks
and the smaller the real effect of nominal shocks, such as monetary policy shocks. In this context,
Golosov and Lucas Jr. (2007) have argued that a strong selection effect generates much smaller
amounts of monetary non-neutrality in menu cost models than in time-dependent pricing models.
Midrigan (2011) challenged this conclusion by showing that Golosov and Lucas’s model did not
match the fraction of price changes that are small as well as the excess kurtosis in the size of price
changes. He then showed that a menu cost model that matches these features could dramatically
decrease the strength of the selection effect, and thereby increase the degree of monetary nonneutrality in menu cost models. In recent work, Alvarez and Lippi (2013) present a model that
3
features menu costs and multi-product firms with an arbitrary number of goods - a generalization
from the two goods in Midrigan (2011). In their model, the initial impact of a monetary shock
becomes smaller and impulse responses become more stretched out, leading to larger real effects,
as the number of goods increases. The broad evidence in this paper confirms the predictions for
several moments of the price distribution implied by Midrigan (2011) and Alvarez and Lippi (2013).
Second, our data provide strong evidence for synchronization of price adjustment decisions
within firms. The extent to which price changes in the economy are synchronized is again a highly
relevant statistic for monetary models because it informs us about the importance of strategic complementarities for pricing decisions. As emphasized recently by Carvalho (2006) and Nakamura and
Steinsson (2008), strategic complementarities can amplify real effects of nominal shocks. To study
synchronization, we estimate a multinomial logit model to relate individual adjustment decisions
to the fraction of price changes of the same sign within a firm, within the same industry, and other
economic fundamentals.
We find that when the price of one good in a firm changes, there is a large increase in the probability that the price of another good in the firm changes in the same direction. A one-standard
deviation increase in the firm fraction of other goods changing price in the same direction is associated with a 14.75 percentage point higher probability of upwards and a 8.82 percentage point higher
probability of downwards price adjustment. Moreover, our results show that such synchronization
within the firm is much stronger than within the industry. We also document that the number
of goods and the degree of within-firm synchronization strongly interact in determining individual
price adjustment decisions. In particular, we find that the strength of within-firm synchronization
increases monotonically with the number of goods. Again, this result holds both for upwards and
downwards adjustment decisions. At the same time, we find that the strength of synchronization
within the same industry decreases monotonically as the number of goods increases. This finding
is complementary to the work of Boivin et al. (2009), as it locates the incidence of pricing decisions
at the level of multi-product firms relative to their industries.
Next, on the theoretical side, we develop a state-dependent pricing model of multi-product firms.
We find that varying only one parameter–the firm-specific menu cost– allows us to qualitatively
4
match all the trends observed in the data. Our modeling approach follows the recent literature, for
example Nakamura and Steinsson (2008), Gopinath et al. (2010), Gopinath and Itskhoki (2010),
and Neiman (2011). Firms in our model face idiosyncratic productivity shocks and an aggregate
inflation shock. The productivity shocks are correlated across goods produced by the same firm.
Moreover, there is a menu cost of changing prices which is firm-specific and there are economies of
scope in the menu cost technology.
The main theoretical results of the model are driven by economies of scope in the menu cost
technology: firms will change prices of some products even though prices of these products are not
very far out of line from the desired price because the prices of other products need to be changed,
and the firm might as well change the prices of all products since it is changing some already. As
the number of goods increases, this mechanism directly leads to a higher frequency of price changes
and a lower mean absolute, positive, and negative size of price changes. It also implies that the
fraction of small price changes is higher for multiproduct firms. Moreover, with trend inflation,
firms adjust downwards only when they receive substantial negative productivity shocks. Since the
firm adjusts prices when the desired price of one item is very far from its current price, a higher
fraction of downward price changes becomes sustainable.
The model also delivers synchronization of price changes: when we compare a 2-good to a 3good firm, the model predicts that both positive and negative adjustment decisions become more
synchronized within the firm. This effect is again due to the underlying economies of scope in
the menu cost technology: they increase the probability of simultaneous adjustment decisions. In
particular, positive adjustment decisions become more synchronized than negative ones due to
shocks from upward trend inflation as the number of goods increases. Finally, correlation of shocks
among goods within a firm amplifies the synchronization of adjustment decisions.
Our empirical work is related to the recent literature that has analyzed micro-data underlying
aggregate price indices.1 Our paper contributes the first account of price-setting dynamics from
the perspective of the firm. Two other recent papers have also used the same data to uncover
interesting patterns. Nakamura and Steinsson (2008) show that there is substantial heterogeneity
1
For a survey of this literature, see Klenow and Malin (2010). Most of this literature has focused on the U.S. or
the Euro Area. For an analysis of emerging markets, see Gagnon (2009).
5
across sectors in the PPI in the frequency of price changes. Matching groups between CPI and
PPI data, they also find that the correlation in the frequency of price changes between the groups
is quite high. Goldberg and Hellerstein (2009) document that price rigidity in finished producer
goods is roughly the same as in consumer prices including sales and that large firms change prices
more frequently and by smaller amounts compared to small firms. Our results are complementary
to theirs. Neither of these papers however, contain a systematic analysis from the perspective of
the firm, and in particular, how price-setting dynamics differ according to the number of goods
produced by firms. Our empirical results are also related to findings from papers that use analyze
pricing behavior by retail or grocery stores, such as Lach and Tsiddon (1996), Fisher and Konieczny
(2000) or Lach and Tsiddon (2007). Moreover, Midrigan (2011) provides some empirical support
for his model based on scanner data from a chain of grocery stores in Chicago (Dominick’s). One
contribution of our paper is to provide much more broad-based empirical evidence supporting
exactly the types of modeling features that Midrigan (2011) emphasized.
On the theoretical side, our model is related to work by Sheshinski and Weiss (1992), Midrigan
(2011), and Alvarez and Lippi (2013). Our main contribution relative to the Sheshinski and Weiss
(1992) and Midrigan (2011) is to consider and systematically analyze price-setting as we vary the
number of goods produced by firms, from 1 to 3 goods, going beyond the case of a 2-good firm.
Thus, we are able to study trends in some price setting statistics like synchronization by considering
2-good vs. 3-good firms which is not possible by only comparing 1-good and 2-good firms.
Alvarez and Lippi (2013) use stochastic control methods to characterize the price setting solution
of a multi-product firm producing an arbitrary number of goods. They analytically show that
given firm-specific menu costs, frequency of price change increases, absolute size decreases, and
the dispersion increases as the number of goods produced increases. We show these same trends
numerically in a slightly different environment. Our relative contribution is that we solve a model
with stochastic inflation and also allow idiosyncratic shocks across goods produced by the same
firm to be correlated. Importantly, we can also generate additional predictions for the direction
and synchronization of price changes. We then match all the empirical trends in these moments
qualitatively using a calibrated model.
6
2
U.S. Producer Price Data and Multi-Product Firms
We use monthly producer price micro-data from the dataset that is normally used to compute the
PPI by the BLS. While we refer the reader for details to the appendix and the existing descriptions
of the PPI data in Nakamura and Steinsson (2008) and Goldberg and Hellerstein (2009), we describe
a key feature of the data in the next subsection that is relevant to our analysis of pricing by multiproduct firms: there is substantial variation in the number of goods across more than 28, 000 firms.
This allows us to study how key pricing statistics vary with the number of goods.
We also note that we use PPI instead of CPI micro data mainly since we have in mind a model
where actual producing firms, not retailers such as supermarkets or grocery stores, set prices. A
similar analysis of producer pricing decisions is not feasible with CPI data since the CPI sampling
procedure does not map to the production structure of the economy. Instead, it maps to stores,
so-called “outlets,” which may sell goods from any number of firms, including imports. This makes
pricing a complicated web of decisions that involves the whole distribution network. Moreover, it
is generally also not possible to identify the producing firms for specific CPI items. The CPI data
only sometimes record the manufacturer of a good. On a technical level, there is also too little
variation in the number of goods in the CPI data to perform an analysis like for the PPI as we
discuss below.
2.1
Identifying and Grouping Multi-Product Firms
We pin down the multi-product dimension of firms by using firm identifiers, and then simply
computing the average number of goods from a firm over the time periods when the firm has at
least one good in the data. We find that the median (mean) number of goods per firm is 4 (4.13)
with a standard deviation of 2.55 goods. The minimum number of goods is 1, the maximum is
77 goods. We use this variation to group firms into four bins according to the average number of
goods that firms have while in the data: a) bin 1: firms with 1 to 3 goods, b) bin 2: firms with
more than 3 to 5 goods, c) bin 3: firms with more than 5 to 7 goods, and d) bin 4: firms with
more than 7 goods. Thus, firms in higher bins sell a greater number of goods than firms in lower
bins. A similar strategy of binning has also been used in Gopinath and Itskhoki (2010), who look
7
at the relationship between frequency of price changes and exchange rate pass-through. Note that
we have many firms who have non-integer numbers of goods due to the averaging of the monthly
number of goods for each firm. This is another reason to have bins.
We summarize the variation in the number of goods in our data in more detail in Table 1. The
median (mean) number of goods per firm across by bins is 2 (2.2), 4.0 (4.0), 6.0 (6.1), and 8.0 (10.3)
respectively. The dispersion in the number of goods is higher in bin 4, with for example, a standard
error of 0.11. The table also shows that while the majority of firms, around 80%, fall in bins 1
and 2, there are a substantial number of firms in bins 3 and 4 as well. In fact, since firms in bins
3 and 4 set more prices per firm, they account for a much larger share of all prices in the data.
Firms in bins 3 and 4 set around 40% of all prices in our data. We verify that the distribution of
the number of goods is not systematically driven by firm size: there is no clear trend in terms of
median employment per good across these different bins as Table 1 shows.2
While the number of goods per firm in the data due to sampling does not map one to one into
the actual number of goods priced by firms, it is important to emphasize that there is a monotonic
relationship between the number of goods in our data and the actual number of goods per firm.
On the one hand, this is due to the BLS sampling procedure. The BLS sampling design in the
“disaggregation” stage is such that all the economically important products tend to be sampled
with probability proportional to their sales.3 In addition, the BLS pays special attention to cover all
distinct product categories if they exist in a firm allowing some discretion in sampling when there
are many products in a firm. Thus, if a firm has more products, more products will be sampled on
average.
On the other hand, our strategy of binning goods into the ranges leaves some room for potential
errors of sampling into the “wrong” bin and allows us to average out such errors when we calculate
our statistics of interest. When at the cut-off to an adjacent bin, the BLS might randomly sample
one good more or less than is representative according to the sampling scheme. As long as this
2
We include detailed controls for firm size later in the paper to check the robustness of our results.
We know from Bernard et al. (2010) and Goldberg et al. (2008)’s Table 4 that large firms are multi-product
firms with substantial value of sales concentrated in a few goods. We present an analogous table for our dataset in
APPENDIX C. Results suggest that sampling is likely to monotonically capture the actual number of economically
important goods. Please see the appendix for details.
3
8
happens randomly, our statistics of interest should still be representative of firms with more goods
as we move to higher bins. Our empirical results also validate our approach of binning: our choice
of binning leads to results that our theoretical model in all cases predicts would be indeed identified
with an increasing number of goods per firm.
Finally, it is worth noting at the outset that sales prices are not prevalent in the PPI unlike in
the CPI data, as documented by Nakamura and Steinsson (2008). Therefore, our analysis does not
distinguish between sale and non-sale prices. However, we do check for the importance of product
substitutions. We can identify product replacement by changes in the so-called “base price” which
contains the price at resampling of a good. When this base price changes within a price time series,
but the data show no change in the actual price series, we set our product substitution dummy to
one. Results remain the same when taking product replacement into consideration. Therefore, all
our results reported below exclude substitutions.
3
Pricing Behavior of Multiproduct Firms
In this section, we show how pricing by multi-product firms does not resemble at all pricing of
multiple single-product firms considered as one unit. This has important consequences for modeling
price setting in macroeconomic models with menu costs, as argued by Midrigan (2011) and Alvarez
and Lippi (2013). We show this result by documenting how key price change statistics have clear
trends in the number of goods per firm. We also show that there is substantial synchronization of
price changes within a firm. Related to work by Boivin et al. (2009), this has important implications
for locating the incidence of shocks and for thinking about strategic complementarities in pricing.
3.1
Aggregate Price Change Statistics
Frequency of price changes
One key statistic of price setting that turns out to be a function of the number of goods is the
frequency of price changes. It is an important statistic because it reflects the extent of nominal
frictions, which are one essential ingredient for generating real effects in New-Keynesian models. It
is also an important calibration target in multi-product menu cost models such as Midrigan (2011)
9
or Alvarez and Lippi (2013).
We compute the frequency of price changes separately for each bin. This allows us to trace
out how it is a function of the number of goods per firm. We compute the frequency in each bin
as the fraction of price changes for a representative good of a representative firm. That is, after
computing the fraction of all monthly price changes over the life-time of a single good, we calculate
the median frequency over all goods within a firm. This gives us one number per firm. Then, we
report the mean, median, and standard error of frequencies across firms in a given bin. We use this
standard error to compute 95% confidence intervals throughout the paper.4 We compute upper
and lower bounds of the bands as ±1.96 ∗ std. error.
We find strong evidence that the frequency of price changes is a function of the number of goods
per firm. Figures 1 and 2 show this graphically: the mean (median) frequency of monthly price
changes increases with the number of goods per firm. The mean frequency increases from 20% in
bin 1 to 29% in bin 4 while the median frequency increases from 15% to 23%. The relationship is
monotonic across bins except for the mean frequency of price changes for bins 1 and 2.5 Clearly,
these trends show that multi-product firms are not the same as an aggregate of multiple singleproduct firms. Finding a large fraction of negative price changes also reaffirms the result from
the literature in Golosov and Lucas Jr. (2007), Nakamura and Steinsson (2008) and Klenow and
Kryvtsov (2008) that models that rely on only aggregate shocks, and hence predict predominantly
positive price changes with modest inflation, are inconsistent with micro data.
Size and distribution of price changes
Further, as Midrigan (2011) and Alvarez and Lippi (2013) argue, a distinction between multiand single-product firms matters because the selection effect works differently for multi-product
firms and changes the distribution of price changes in response to shocks. We observe exactly this
4
In this exercise, we do not count the first observation as a price change and assume that a price has not changed if
a value is missing, following Nakamura and Steinsson (2008). We have also verified that left-censoring of price-spells
is not a problem, and that our trends are the same when we take means across goods at the firm level.
5
One can interpret the frequency not only as the monthly probability of a price change, but also in terms of the
duration of price spells. Inverting these frequency estimates implies that the mean duration of a price spell decreases
from 5 months in bin 1 to 3.4 months in bin 4 while the median duration decreases from 6.7 months to 4.3 months.
These results are similar to those found for example by Goldberg and Hellerstein (2009). We also note that 64%
of these changes are positive price changes in bin 1 going down to 61% in bin 4, as one would expect under trend
inflation, and as summarized in Figure 3.
10
relationship between various measures of the distribution of price changes and the number of goods
in our data: firms with more goods change their prices by smaller amounts, price changes are more
dispersed and there are more small price changes.
First, we consider the absolute size of log price changes as a measure of the magnitude of price
changes, conditional on adjustment. When we aggregate from the good to the firm to the bin
level as before, we find that the absolute size of price changes decreases with the number of goods
produced by firms as it goes down monotonically from 8.5% in bin 1 to 6.6% in bin 4. Figure 4
shows this graphically. This trend is a key moment for the calibration of the menu cost model later
in our paper, and in Alvarez and Lippi (2013).
This relationship holds even when we separate out the price changes into positive and negative
price changes, conditional on adjustment. Figure 5 shows that the size of positive price changes
decreases with the number of goods while the size of negative price changes increases with the
number of goods. Thus, in general, firms with a greater number of goods adjust their prices by
smaller amounts, both upwards and downwards.
Second, we consider two other statistics that describe the distribution of price changes and that
are key to the selection results in Midrigan (2011) and Alvarez and Lippi (2013): the fraction of
“small” price changes and the kurtosis of the distribution. We define a price change as small if
|∆pi,j,t | ≤ κ|∆pi,t |, where i is a good in firm j, and κ = 0.5, following Midrigan (2011). That is,
a price change is small if it is less in absolute terms than a specified fraction of the mean absolute
price change in a firm. After creating an indicator variable, we report the fraction of small price
changes for each bin.
We find – just as menu cost models of Midrigan (2011) and Alvarez and Lippi (2013) that
generate large real effects of monetary shocks predict – that small price changes are quite prevalent.
Figure 6 shows that the fraction of small price changes increases from 38% in bin 1 to 55% in bin
4. Thus, small price changes also become more prevalent when firms produce many goods. Our
empirical findings mirror the results in Klenow and Kryvtsov (2008) who report that 40% of price
changes in the U.S. CPI data are smaller in absolute terms than 5%.
Our analysis also provides broad-based empirical evidence consistent with a leptokurtic distri-
11
bution of shocks that can generate large real effects of monetary shocks as posited by Midrigan
(2011) and Alvarez and Lippi (2013). We provide such evidence by computing the kurtosis of price
changes, which is the ratio of the fourth moment about the mean and the variance squared:
n
T
i
t=1
i
1 XX
µ4
K = 4 where µ4 =
(∆pi,j,t − ∆pj )4
σ
T −1
(1)
where in a given firm j with n goods, ∆pi,t denotes the log price change of good i, ∆p the mean
price change and σ 4 is the square of the usual variance estimate. We find strong evidence that price
changes are leptokurtically distributed. Again, the extent of this is a function of the multi-product
nature of firms. Figure 7 shows that the mean kurtosis of price changes increases with the number
of goods produced by firms as it goes up from 5.3 in bin 1 to 16.8 in bin 4. It is worthwhile noting
that the model in Midrigan (2011) generates large real effects exactly when there is excess kurtosis,
that is, kurtosis larger than 3. Consistently with a more dispersed price change distribution for
multi product firms, we also document in Figure 8 that both the 1st and the 99th percentiles of
price changes take more extreme values as the number of goods increases.
Finally, the work by Alvarez and Lippi (2013) makes clear predictions for the coefficient of
variation of absolute price changes as the number of goods increases. Again, we find broad-based
support for the prediction of their multi-product menu cost model. There are strong trends in the
coefficient of variation suggesting that multi-product firms price differently than an aggregate of
single-product firms. To compute the coefficient, we pool our data at the firm level. Then, we
compute the ratio of the standard deviation to the mean of absolute price changes for an item,
take the mean across items in a firm, and then means across firms in a bin. Table 2 shows that
the coefficient of variation monotonically increases with the number of goods, from 1.02 in bin 1 to
1.55 in bin 4.
Robustness
Our aggregate results above are completely robust to controlling for various factors that might
lead to trends across bins. In particular, we show that it is not heterogeneity in terms of firm
size, substitutability or sectoral characteristics that drives our trends. It is also not the case that
12
the large fraction of small price changes is an artifact of the data, as argued by Eichenbaum et al.
(2013). All the results in this section can be found in the online Appendix A.
First, we show that the different goods bins are not picking up effects of differential firm size.
We demonstrate this for the sake of conciseness only for the frequency and size of price changes.
When we control linearly for the size of firms in a regression, using the total number of employees
per firm as a measure of size, we find that our trends across bins continue to hold. Figures A.1 and
A.2 show this result for the frequency and size of price changes.
Next, we show in several steps that various kinds of heterogeneity across bins does not affect our
results. As a first pass, we present the distribution of firms across bins and industries at the 2-digit
NAICS level. We find that no particular industry substantially dominates a bin, as summarized
in Table A.1. Most firms belong to the manufacturing sector, and typically bins 1 and 2 contain
the vast majority of firms. Notable exceptions are NAICS 22 (utilities) which contains a very high
proportion of firms that fall in bin 4, and NAICS 62 (health care and social assistance) where
almost half the firms are in bins 3 and 4. This broadly flat composition across industries also holds
at more disaggregated levels, as shown in Tables A.2 and A.3.
As a more rigorous test of whether heterogeneity is the driving force of our results, we show
that the same price-setting trends across bins hold even after controlling for industry fixed effects
at 2-, 3-, and 4-digit NAICS sectoral levels. Figures A.3 and A.4 summarize these results. Since
a majority of firms are in manufacturing, it is also natural to wonder if our results are valid only
for these sectors. This is not the case as we show in Figures A.5 to A.8. The trends in frequency
and size remain whether we take out manufacturing sectors or whether we compute these statistics
separately for each two-digit manufacturing sector.
As a final step to account for the effect of heterogeneity through regression, we explicitly use
a set of detailed fixed effects as a filter. That is, we filter out month-, product-, and firm-level
fixed effects from price changes, with product-level fixed effects defined at the 4-, 6- and 8-digit
PPI product codes. Then, we ask how much in the variation these fixed effects can explain. We
13
estimate the following specification regarding variation in |∆p|, the absolute size of price changes:
|∆p|i,f,p,t = α0 {DM onth
m m=12
}m=1
P roduct
F irm
+ α1 Di,f,p,t
+ α2 Di,f,p
+ i,f,p,t
(2)
where i denotes an item, f a firm, p a product at the 4-, 6-, and 8-digit level, and t time. Dummies
are for months, products, and firms. This is a standard decomposition similar to the one performed
in Midrigan (2011). We show in Table A.4 that the variation in price changes explained by these
fixed effects is at most 29%.
Importantly, trends across bins are also not due to a particular kind of heterogeneity: varying
elasticities of demand or degrees of substitution. We find that the positive relationship between
frequency of price changes and number of goods, and the negative relationship between size of price
changes and number of goods continue to hold even when we tightly control for varying elasticities
of demand or substitution. We verify this by first picking firms that sell goods in a narrow product
code provided by the BLS at the 6-digit level, excluding firms which sell in multiple product codes.
Then, we compute the median fraction of price changes, the absolute size of price changes, and
the number of goods sold by the firms at a point in time. Third, we run two regressions: first,
we regress the frequency of price changes on the number of goods and second, the absolute size of
price changes on the number of goods. We take the median of the estimated coefficients across all
product codes, and then report summary statistics of these medians over time. Tables A.5 and A.6
present our results.
Finally, we verify in detail that our results on small price changes are robust to various alternative definitions, as well as the concerns brought forward in recent work by Eichenbaum et al.
(2013). This is particularly relevant for the aggregate theoretical results of Midrigan (2011) and
Alvarez and Lippi (2013). Recall that in our baseline results, we defined small price change as
|∆pi,j,t | ≤ κ|∆pi,t | where i is a good of firm j and κ = 0.5. In Table A.7 we show that our results
continue to hold for κ = 0.10, 0.25, and 0.33. Our results also continue to hold if we define a price
change as small relative to the industry mean, or relative to the good level mean. Table A.7 also
shows that our results are strongly robust if we define small price changes in terms of absolute
values. That is, price changes that are less than 0.25%, or 0.5%, or 1%.
14
Our findings are robust to the recent concerns brought forward by Eichenbaum et al. (2013):
measurement error, for example induced by bundling or uncontrolled forms of quality changes,
could falsely generate many small price changes. However, we find that this is not a problem for
our PPI data. When we exclude the six-digit NAICS sectors that likely encompass the problematic
ELIs from the CPI identified by Eichenbaum et al. (2013), our overall, estimated fraction of small
price changes remains essentially unchanged across definitions. The reason is that many of the
ELIs that may be problematic in the CPI are not in the set of manufactured producer goods which
are the bulk of our data. Additionally, we try focusing only on price changes that are bigger than
one penny, or larger than 0.01%, or 0.1%. This leaves our results mostly unchanged even though
the overall fraction somewhat falls by up to 3.5%.
3.2
Adjustment Decisions and Synchronization
To further improve our understanding of individual pricing decisions, we go beyond providing aggregate statistics, and estimate a discrete choice model of pricing decisions. We find some important
results: First, there is substantial synchronization of price changes within a firm which suggests
a vital role played by strategic complementarities in price-setting. Second, there is substantial
synchronization of individual adjustment decisions at the firm level relative to the industry, with
strong trends as the number of goods increases. This finding confirms that multi-product firms
do not behave like a collection of single-product firms, while it is also complementary to the work
of Boivin et al. (2009) in locating the incidence of individual pricing decisions. Third, we find
evidence for elements of state-dependent pricing in response to fundamental economic variables.
First, results show substantial synchronization of individual adjustment decisions at the firm
level and especially relative to the industry, with strong trends as the number of goods increases.
To arrive at this result, we estimate a multinomial logic model of the following form:
P r(Yi,j,t = 1, 0, −1|Xi,j,t = x) = Φ(βXi,j,t )
(3)
where Yi,j,t is an indicator variable for upwards, no, or downwards price changes of good i produced
by firm j at time t, with 0 as the base category. We use estimates of β to report marginal effects on
15
the change in the probability of adjustment, given one-standard-deviation changes around the mean
of Xi,j,t . We focus on estimation separately by bins which distills out trends of how multi-product
firms are different from single-product firms. However, since the results for the PPI as a whole are
of independent interest, we estimate the model on pooled data across all good bins as well.
Our explanatory variables Xi,j,t include three sets of variables: First, we try to capture the
extent of synchronization in price setting at the firm and the sectoral level. To this purpose, we use
the fractions of same-signed price changes within the firm, and the same six-digit NAICS sector,
excluding the price change of the good we are trying to explain. Second, we include a dummy for
product replacement where we can identify it by changes in the base price at resampling. Finally,
we control for energy and food-inclusive CPIs, the number of employees in the firm, industry fixed
effects, month fixed effects, time trends and as a measure of marginal costs, we also include the
average price change of goods in the same firm and six-digit NAICS sector. We supplement our
data with monthly inflation rates from the OECD “Main Economic Indicators (MEI).” We use
both CPI inflation including food and energy prices, as well as excluding food and energy prices.
Since we find no qualitative difference in our subsequent results, we only report results from the
inclusive CPI measure.
Synchronization of price changes We find that there is pervasive synchronization both at
the firm level and the industry level, exhibiting clear trends. The strongest synchronization is at
the level of the firm: When the fraction of price changes of the same sign in a firm changes around
its mean by one standard deviation, the pooled data tell us that the probability of a downward price
change of a good increases by 8.82 percentage points, while it increases for an upward price change
by 14.73 percentage points.6 Table 3 shows this result, as well as the strong trends across bins: as
the number of goods increases, the statistical and economic importance for adjustment decisions
of what happens within a firm monotonically increases. When the fraction of price changes of
the same sign changes by one standard deviation around the mean, the effect goes up from 5.38
percentage points to 15.78 percentage points for negative price changes, and from 9.61 percentage
points to 24.24 percentage points for positive price changes.
6
To avoid cluttering, we do not report p-values, but all the results we report are statistically significant.
16
These large complementarities in adjustment decisions within firms plus the strong trends underline our message that multi-product firms price differently than a collection of single-product
firms. The trends also suggest that the importance of firm-specific shocks becomes more important
as the number of goods increases. This is related to work by Boivin et al. (2009). Comparing the
explanatory power from a regression with and without firm-specific variables also emphasizes their
importance: when we omit the firm-specific fraction and the average price change of goods from
an otherwise unchanged regression, R2 goes down from 48% to 30%. Curiously, the R2 also shows
a positive trend with the number of goods.
The importance of the multi-product dimension for pricing is further reinforced when we consider synchronization with decisions in the industry: There, we find that synchronization is much
weaker, especially as the number of goods increases. On average, the probability of a downward
price change of a good increases by 1.32 percentage points, while the probability of an upward
price change of a good increases by 2.27 percentage points when the industry-level fraction of price
changes of the same sign moves around the mean by one standard deviation. As the number of
goods increases, the effect goes down from 3.07 percentage points to 0.21 percentage points for
positive price changes, and from 2.10 percentage points to −1.51 percentage points for negative
price changes.
Here, it is important to emphasize that our synchronization results are not due to a purely
“statistical” effect. One could think that the variance of the fraction of price changes of the same
sign in a firm decreases as the number of goods increases. This could imply a larger estimated
synchronization coefficient. However, there is no such trend in the data for the variance of fractions.
In fact, the opposite holds true.
State-dependent response to inflation
It is a long-debated and important question in
monetary economics whether price setting is time-dependent or state-dependent. We contribute
some evidence by explicitly considering the role played by inflation, an important aggregate shock,
for pricing decisions. We find what one would expect from a model where firms adjust prices in
a state-dependent fashion. As Table 3 shows, the likelihood of a price decrease decreases with
higher CPI inflation while the likelihood of a price increase increases. When CPI inflation changes
17
by one standard deviation around the mean, the probability of a negative price change of a good
decreases by 0.21 percentage points, while the probability of a positive price change increases by
0.17 percentage points. The effects are both statistically and economically significant.
Robustness We conduct a battery of robustness tests for these good-level regressions. First,
we consider different levels of aggregation for our definition of the industry variable. In conducting
this extension, in addition to checking for robustness, we are motivated by the theoretical results
in Bhaskar (2002) that synchronization is likely to be higher within groups with higher elasticity of
substitution among goods, that is, at a more disaggregated industry level. Table A.8 in Appendix
A presents results using an industry classification at the 2-digit NAICS level. It shows that at this
higher level of aggregation, prices in the pooled data specification are much less synchronized at the
industry level, compared to Table 3. Table A.8 also shows results for the four good bins separately.
The general result of higher firm level synchronization and lower industry level synchronization as
we move to higher good bins is robust to this alternate definition of industry.7 Compared to Table
3 and as predicted by Bhaskar (2002), however, we see the significant extent to which the industry
level synchronization has decreased across all bins.
Second, we check that clustering standard errors at the industry level do not affect out findings.
Third, we use a polynomial function for the size of firms to control for non-linear size effects in the
regressions. Our results do not change due to this modification. Fourth, we use a CPI measure
that excludes food and energy prices. Finally, we estimate the multinomial logit model only for the
adjustment decisions for the largest sales-value item of each firm. Again, our results do not change
due to this modification, in particular, with respect to synchronization.8
7
This result is similar to that of Cornille and Dossche (2008) and Dhyne and Konieczny (2007) who use the Belgian
PPI and the CPI respectively. They use the Fisher-Konieczny measure of synchronization and find that prices tend
to be more synchronized at a more disaggregated industry level. Neither of these papers look at the level of the firm,
however, which is the focus of our paper, and also the factor that quantitatively matters most in our data.
8
The results mentioned in the second robustness paragraph are available upon request from the authors.
18
4
Theory
Compared to the existing literature, our main theoretical challenge is to explain the various trends
we observe in price setting as we vary the number of goods per firm, an analysis that has not been
undertaken before. We do so by developing a model of pricing by state-dependent multi-product
firms. We find that varying only one parameter – the firm-specific menu cost of price adjustment –
allows us to match the trends in the data. Since the mapping from the good bins that we construct
in the empirical section to the number of goods in the model is not direct, we view our exercise in
this section as qualitative in nature. Nonetheless, the results from simulations of our model allow
us to validate modeling features that are needed to explain the empirical trends.
4.1
Model
We use a partial equilibrium setting of a firm that decides each period whether to update the
prices of its n goods indexed by i ∈ (1, 2, 3), and what prices to charge if it updates. Our model
is similar to the ones in Sheshinski and Weiss (1992), Midrigan (2011), and Alvarez and Lippi
(2013). The main difference is that compared to Sheshinski and Weiss (1992) and Alvarez and
Lippi (2013) we allow for a stochastic aggregate shock and correlated idiosyncratic shocks across
goods produced by a firm, while compared to Midrigan (2011), we solve for equilibrium as we vary
n. In particular, the latter variation allows us to make two important contributions. First, we can
solve for trends in price-setting behavior with respect to how many goods firms produce. Second,
we can compute trends in synchronization of price-setting by comparing 2-good and 3-good firms.
Since no measures of synchronization can be computed in the one-good case, the comparison of
2-good and 3-good firms is necessary to model trends in synchronization of adjustment decisions.
This focus on synchronization also additionally distinguishes our model from the aforementioned
papers.
Production technology and demand
In our model, the firm produces output ci,t of good i using a technology that is linear in labor
li,t :
ci,t = Ai,t li,t
19
where Ai,t is a good-specific productivity shock that follows an exogenous process:
ln Ai,t = ρiA ln Ai,t−1 + iA,t
2 for i = j and σ 2
where E[iA,t ] = 0 and cov(Ai,t , Aj,t ) = σA
Ai ,Aj for i 6= j. That is, as in Midrigan
(2011) we allow the productivity shock across goods produced by the same firm to be correlated.
The firm’s product i is subject to the following demand:
ci,t =
pi,t
Pt
−θ
Ct
i = 1, 2, ..., n
where Ct is aggregate consumption, Pt is the aggregate price level, pi,t is the price of good i, and
θ is the elasticity of substitution across goods. In this partial equilibrium setting, we normalize
Ct = C̄. We also assume that the price level Pt exogenously follows a random walk with a drift:
ln Pt = µP + ln Pt−1 + P,t
where E[P,t ] = 0 and var(P,t ) = (σP )2 . Given our assumption about technology, the real marginal
cost of the firm for good i, M Ci,t , is therefore given by: M Ci,t =
wage. We normalize
Wt
Pt
Wt
Ai,t Pt ,
where Wt is the nominal
= w̄.
Given this setup, the firm maximizes the expected discounted sum of profits from selling all of
its goods. Total period gross profits are given by:
πt =
n X
pi,t
i
Pt
−
w
Ai,t
pi,t
Pt
−θ
C̄.
The problem of the firm is to choose whether to update all prices in a given period, and if so, by
how much. Whenever it updates prices, it has to pay a menu cost K, which we discuss next.
Menu cost technology
In particular, this “menu cost” comes in the form of a constant, firm-specific cost K(n) > 0 for
adjusting one, or more than one, of its prices. This assumption of firm-specificity directly implies
that the firm will either adjust all its prices at the same time or adjust none at all. In our dataset,
20
this is a good first-order assumption: conditional on observing at least one adjustment per firm,
the total fraction of goods adjusting in a firm is 0.75.
A key feature of our adjustment technology is economies of scope: The cost of changing prices
depends on the number of goods produced by the firm and in particular, we assume that:
∂K(n)
∂ 2 K(n)
< 0.
> 0 and
∂n
∂n2
These assumptions mean that the cost of changing prices increases monotonically with the number
of goods, and that there are increasing cost savings as more and more goods become subject to
price adjustment. Therefore, there are economies of scope in the cost of changing prices which
will allow us to match the trends in the data. A priori, however, it is unclear if we can match
all observed trends by varying only one parameter K: On the one hand, it is very intuitive that
economies of scope imply smaller and more frequent price changes as the number of goods increases
per firm. On the other hand, other key statistics of the distribution or synchronization could show
any trend.
How should we understand the nature of these “menu costs”? We view them very broadly as a
general fixed cost of price adjustment, not as the literal cost of relabeling the price tags of goods.
Blinder et al. (1998) provides some evidence for this general interpretation. While we do not model
the adjustment process in detail, one can for example think about the adjustment technology as
a fixed cost of paying a manager to change prices: it is costly to hire him in the first place, but
much less costly to have him adjust the price for each additional good. Stella (2011), using a
structural estimation approach, provides evidence that indeed a common component in a multiproduct setting accounts for 21% to 90% of adjustment cost independent of the number of goods,
with the remainder attributable to variable costs. Another way to think about adjustment costs is
to interpret them as a fixed cost of marketing, information gathering and pricing efforts that can
be shared across goods in a firm, as suggested by Zbaracki et al. (2004). We conclude from the
literature that there is empirical evidence supporting our modeling choice. We leave it to future
empirical work to further dissect the underlying micro mechanisms.
21
4.2
4.2.1
Results
Computation
We solve the given problem for a firm that produces n = 1, 2, and 3 goods. First, we employ
collocation methods to find the policy functions of the firm. Second, given the policy functions, we
simulate time series of shocks and corresponding adjustment decisions for many periods. Finally,
we compute statistics of interest for each simulation and good, and across simulations. We also
estimate a multi-nomial logit model of adjustment decisions using the simulated data to compare
our theoretical results with the empirical findings. Appendix B provides further details about
computation and analysis.
We mostly follow the literature in our choice of parameters. Since our model is monthly, we
1
choose a discount rate β of (0.96) 12 . We choose θ to be 4, which implies a markup of 33%. To
parametrize the aggregate exogenous processes, we set the trend in aggregate inflation µP to be
a monthly increase of 0.21% and the standard deviation σP to be 0.32%. These values are taken
from Nakamura and Steinsson (2008). To parametrize the idiosyncratic productivity shock, we
use the NBER productivity database covering 459 sectors from 1984 − 1996 and compute for our
productivity process an estimate of ρ = 0.96 for the AR(1) parameter and σA = 2% for the
standard deviation.9 For the firms with 2 and 3 goods, we use the same values for the persistence
and variance of the all idiosyncratic productivity shocks, but following Midrigan (2011), we allow
for a correlation of 0.65 among the good-specific productivity shocks. Finally, we use a value of
K such that menu costs are a 0.40% of steady-state revenues for the 1-good firm, 0.70% for a 2-good
firm, and 0.80% for a 3-good firm. These choices for K allow us to replicate, as shown below, the
trends across the good bins that we document empirically.
4.2.2
Findings
We find that the model predicts clear and systematic trends in the key price-setting statistics as we
increase the number of goods from 1 to 3. The trends align, qualitatively, with our empirical finding
9
A persistent productivity process is a common specification in the literature. For example, Midrigan (2011) uses
a random walk process for idiosyncratic productivity shocks.
22
that multi-product firms behave systematically different from a collection of multiple single-product
firms. We present the main results from our simulations in Table 4.
First, we find that the frequency of price changes goes up from 13.89% to 20.21% while the mean
absolute size of price changes goes down from 5.42% to 4.18% as we increase the number of goods.
Second, the decrease in the absolute size of price changes also holds for both positive and negative
prices changes: they go down from 5.53% to 4.34% and from −5.24% to −3.95% respectively. Third,
the fraction of small price changes increases from 2.29%, barely none, to 19.54%. Thus, while firms
with more goods change prices more frequently, they do so by smaller amounts on average. Fourth,
we also see that the fraction of positive price changes decreases from 62.63% to 60.31%. Thus, as
in the data, firms with more goods adjust downwards more frequently. Finally, the model predicts
that kurtosis increases from 1.45 to 2.03, again consistent with our empirical findings.
What is the mechanism behind our results? For simplicity, compare a 1-good firm with a 2-good
firm. For the case of a 2-good firm, when the firm decides to pay the firm-specific menu cost to
adjust one of the prices, it also gets to change the price of the other good for free. This leads to
a higher average frequency of price changes. At the same time, for the 2-good firm, since a lot of
price changes happen even when the desired price is not very different from the current price of
the good, the mean absolute size of price changes is lower. This smaller mean also implies that the
fraction of “small” price changes is higher for the 2-good firm. In fact, for the 1-good firm, which
is the standard menu cost model, the fraction of small price changes is negligible because in that
case, the firm adjusts prices only when the desired price is very different from the current price.
What causes the decrease in the fraction of positive price changes? With trend inflation, firms
adjust downwards only when they receive very big negative productivity shocks. With firm-specific
menu costs, since the firm adjusts both prices when the desired price of one good is very far from
its current price, it is now more sustainable to have a higher fraction of downward price changes.
Finally, kurtosis increases as we go from one good to two goods because of a higher fraction of price
changes in the middle of the distribution.
Next, we address trends in synchronization of individual good-level price changes. Using simulated data, we run the same multinominal logit regression as in the empirical section given by
23
specification (3). Our explanatory variables now include the fraction of same-signed adjustment
decisions at time t within the firm and Πt is the inflation rate at time t. It is important to emphasize
here the need to go beyond the 2-good case in Midrigan (2011) or Sheshinski and Weiss (1992), and
consider a 3-good case. Otherwise, we cannot check if the model predicts trends in synchronization
of price changes that are consistent with the empirical findings.
We find from our simulations that the strength of synchronization, that is the coefficient estimate
in the logit regressions for the fraction of other goods of the firm changing in the same direction,
increases as we go from a 2-good firm to a 3-good firm. Table 4 shows this. This effect is first simply
due to the underlying economies of scope: they increase the probability of simultaneous adjustment
decisions in a firm as the number of goods increases. Second, correlated shocks in our model also
play a role for synchronization since, as is intuitive with correlated shocks across goods, desired
prices of goods within the firm become more likely to move in the same direction. Importantly, as
is the case empirically, this is also the case with both upwards and downwards price changes.
In addition, our simulation results make clear that the simulations predict that positive price
adjustment decisions are more synchronized than negative adjustment decisions since the synchronization coefficient for upwards price adjustment decisions is always higher than the coefficient for
downwards adjustment decisions. This difference in synchronization probabilities is due to positive
trend inflation. If we omit positive trend inflation from the model, the difference disappears. The
model thus matches our findings on synchronization established in the empirical section.
4.2.3
Robustness
In this section, we discuss alternative modeling mechanisms to economies of scope in menu costs.
Our main finding is that each alternative mechanism can match some of the trends, but it usually
also fails in matching some other trends. This validates the choice of our main, parsimonious model
while suggesting which other mechanisms may be important depending on other modeling interests.
All detailed results, if not in Appendix B, are available on request.
First, we investigate whether correlated shocks are the most important feature of the model as
opposed to economies of scope in menu costs. When we shut down correlation of productivity shocks
24
by setting σAi ,Aj = 0, in fact we can still match qualitatively all of our trends in pricing statistics.10
We summarize this experiment in Table B.1 in Appendix B. Moreover, we have also simulated a
model with correlation of the productivity shocks but without complementarities in menu costs.
Table B.2 summarizes this modification, showing that correlation alone cannot generate the trends
found in the data. Thus, economies of scope in menu costs are critical to generate the trends in
the model. In particular, just firm-specific menu costs, without economies of scope and absent
correlation, are not enough to replicate the trends from our empirical section. Table B.3 shows the
results from a model specification where menu costs are firm-specific, but the per-good menu cost
is the same across firms with 1, 2, or 3 goods.
Second, we investigate the possibility that perhaps our results are driven by the fact that firms
that produce more goods also produce more substitutable goods. We simulate a specification of 2good and 3-good firms with no economies of scope in the cost of adjusting prices and no correlation
of productivity shocks but with an elasticity of substitution among goods that is higher than that
of the 1 good firm. While matching the trends in frequency, size, and fraction of positive price
changes, we generally find that this specification fails to match trends in the fraction of small price
changes, kurtosis, and synchronization as we increase the number of goods.
Third, there might be concern that our synchronization results could be due to purely mechanical, statistical reasons. To investigate this possibility, we perform two tests. In the first, we run a
Monte-Carlo exercise based on a simple statistical model of price changes. We model price changes
as i.i.d. Bernoulli trials with a fixed probability of success. Using simulated time series for an arbitrary number of goods, we estimate our synchronization equation. We find no mechanical trend
in synchronization. In the second test, we use the single-good firm from our simulation exercise
and model a multi-product firm as a collection of multiple, independent single-product firms. That
is, a 2-good firm now is simply a collection of two single-good firms, with no economies of scope
in cost of adjusting prices and no correlated shocks. Similarly, a 3-good firm is a collection of
three independent single-good firms. When we run our synchronization test with simulated data
from these specifications, we find that the model cannot produce synchronization results that are
10
It is still worthwhile to include this feature in the baseline model since it is intuitive, has been used frequently in
the literature, and does generate higher synchronization of price changes.
25
consistent with the empirical findings. For example, the synchronization coefficients estimated using simulated data become smaller as we move from two single-good to three single-good firms.
Incidentally, this specification of multiproduct firms as multiple single-good firms fails to match
any of our aggregate trends in price-setting, such as the frequency of price changes. We summarize
results in Table B.4 in Appendix B. These results highlight the need to model a multi-product firm
as distinctly different from an aggregate of multiple single-product firms.
Fourth, we consider two alternative demand specifications. In the first specification, we allow for
non-zero cross-elasticities of demand among the goods per firm, which is precluded in our baseline
model with CES demand. Again, we shut down economies of scope and correlation of productivity
shocks in our experiments. While this experiment generates a higher fraction of small price changes
and greater kurtosis, we find that it cannot account for the trend in the size of price changes. In
the second specification, we allow for idiosyncratic demand shocks as an alternative to productivity
shocks. We find that demand shocks are unable to generate the fraction of negative price changes
that we observe in the data. The intuition for this result is that large negative demand shocks
increase the menu costs relative to current profits, thus making negative price changes too costly.
5
Conclusion
In this paper, we have established three new facts regarding multi-product price-setting in the U.S.
Producer Price Index. First, we show that as the number of goods increases, price changes are
more frequent, the size price changes is lower, the fraction of positive price changes decreases, and
price changes become more dispersed. Second, we find evidence for substantial synchronization of
price adjustment decisions within the firm. Third, we find that the number of goods and the degree
of synchronization within firms strongly interact in determining price adjustment decisions: as the
number of goods increases, synchronization within firms increases.
Motivated by these findings, we present a model with firm-specific menu costs where firms are
subject to both idiosyncratic and aggregate shocks. We show that as we change the number of goods
produced by the firms, the patterns predicted by the model regarding frequency, size, direction,
dispersion, and synchronization of price changes are consistent with the empirical findings.
26
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“Menu Costs, Multi-Product Firms, and Aggregate Fluctuations,”
Miranda, M. J. and P. L. Fackler (2002): Applied Computational Economics and Finance,
MIT Press.
Nakamura, E. and J. Steinsson (2008): “Five Facts about Prices: A Reevaluation of Menu
Cost Models,” Quarterly Journal of Economics, 123, 1415–1464.
Neiman, B. (2011): “A State-Dependent Model of Intermediate Goods Pricing,” Journal of
International Economics, 85, 1–13.
Sheshinski, E. and Y. Weiss (1992): “Staggered and Synchronized Price Policies under Inflation:
The Multiproduct Monopoly Case,” Review of Economic Studies, 59, 331–59.
Stella, A. (2011): “The Magnitude of Menu Costs: A Structural Estimation,” Working paper,
Federal Reserve Board.
Zbaracki, M. J., M. Ritson, D. Levy, S. Dutta, and M. Bergen (2004): “Managerial and
Customer Costs of Price Adjustment: Direct Evidence from Industrial Markets,” The Review of
Economics and Statistics, 86, 514–533.
28
6
Tables
Table 1: Summary Statistics by Number of Goods
Mean Employment
Median Employment
% of Prices
Mean # of Goods
Std. Error # Goods
Std. Dev. # Goods
Minimum # of Goods
Maximum # of Goods
25% Percentile # Goods
Median # Goods
75% Percentile # Goods
Number of Firms
1-3
2996
427
17.15
2.21
0.01
0.77
1
3
1.78
2
3
9111
Number of Goods
3-5
5-7
>7
1427 1132 1016
155
195
296
43.53 18.16 21.16
4.05
6.06 10.26
0.00
0.01
0.11
0.34
0.40
4.88
3.01
5.01
7.02
5
7
77
3.94
5.91
7.98
4
6
8
4
6
11.77
13577 3532 2160
We compute statistics from the PPI data according the number of
goods. We calculate mean and median employment by taking means
and medians of the number of employees per good across firms in a
bin/overall category. % of Prices denotes the fraction of prices in the
PPI set by firms in each bin.
Table 2: Coefficient of Variation of Price Changes
Firm-Based
Good-Based
1-3
1.02
(0.01)
0.96
(0.01)
Number of Goods
3-5
5-7
1.15
1.30
(0.01) (0.02)
1.00
1.10
(0.01) (0.02)
>7
1.55
(0.02)
1.24
(0.02)
For the first column, we compute the coefficient of variation at the level of the firm pooling all price changes.
Then, we take medians across firms, bin by bin. For the
second column, we compute the coefficient of variation for
each good, take the median across goods within the firm
and then medians across firms.
29
Table 3: Marginal Effects by Number of Goods, ± 1/2 Std. Dev.
1-3 Goods
3-5 Goods
5-7 Goods
>7 Goods
All Goods
All Goods†
2.10%
5.38%
-0.08%
1.41%
6.52%
-0.14%
0.89%
9.92%
-0.17%
-1.51%
15.78%
-0.54%
1.32%
8.82%
-0.21%
6%
-0.25%
3.07%
9.61%
0.17%
42.85%
2.12%
11.46%
0.10%
47.93%
1.55%
15.94%
0.14%
48.33%
0.21%
24.24%
0.35%
49.10%
2.27%
14.74%
0.17%
47.56%
10.29%
0.16%
29.33%
Negative Change
Fraction Industry
Fraction Firm
πCP I
Positive Change
Fraction Industry
Fraction Firm
πCP I
R2
The table shows the bin-specific marginal effects in percentage points of a one-standard deviation change in the
explanators around the mean on the probability of adjusting prices upwards or downwards. Marginal effects are
calculated for a model estimated for each bin separately as well as on the pooled data. All reported effects are
statistically significantly different from zero. †Results from omitting firm-specific variables are shown in the last
column.
Table 4: Results of Simulation
Frequency of price changes
Size of absolute price changes
Size of positive price changes
Size of negative price changes
Fraction of positive price changes
Fraction of small price changes
Kurtosis
1st Percentile
99th Percentile
1 Good
13.89%
5.42%
5.53%
-5.26%
62.92%
2.41%
1.45
-7.24%
7.45%
2 Goods
17.85%
4.60%
4.66%
-4.52%
62.03%
15.92%
1.72
-7.94%
8.36%
3 Goods
20.21%
4.18%
4.34%
-3.95%
60.31%
19.54%
2.03
-8.68%
9.04%
Synchronization measures:
Fraction, Upwards Adjustments
Fraction, Downwards Adjustments
Correlation coefficient
Menu Cost
0.40%
31.29
30.72
0.65
0.70%
38.65
37.72
0.65
0.80%
We perform stochastic simulation of our model in the 1-good, 2-good and 3-good
cases and record price adjustment decisions in each case, allowing for correlation of
the productivity shocks Ai,t in the multi-good cases. Then, we calculate statistics
for each case as described in the text. In the 2-good and the 3-good cases, we report
the mean of the good-specific statistics. We obtain the synchronization measure
from a multinomial logit regression analogous to the empirical multinomial logit
regression. We control for inflation. Menu costs are given as a percentage of steady
state revenues.
30
7
Graphs
Figure 1: Mean Frequency of Price Changes with 95% Bands
Mean Frequency of Price Changes with 95% Bands
31%
Frequency of price changes
29%
27%
25%
23%
21%
19%
17%
15%
1-3
3-5
5-7
>7
Number of goods
Based on the PPI data we group firms by the number of goods. We compute the mean frequency of price changes
in these groups in the following way. First, we compute the frequency of price change at the good level. Then,
we compute the median frequency of price changes across goods at the firm level. Finally, we report the mean
across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96 * std. error
across firms.
31
Figure 2: Median Frequency of Price Changes
Median Frequency of Price Changes
24%
Median frequency of price changes
22%
20%
18%
16%
14%
12%
10%
1-3
3-5
5-7
>7
Number of goods
Based on the PPI data we group firms by the number of goods. We compute the median frequency of price
changes in these groups in the following way. First, we compute the frequency of price change at the good level.
Then, we compute the median frequency of price changes across goods at the firm level. Finally, we report the
median across firms in a given group.
32
Figure 3: Mean Fraction of Positive Price Changes with 95% Bands
Mean Frequency of Positive Price Changes with 95% Bands
66%
Fraction of positive price changes
65%
65%
64%
64%
63%
63%
62%
62%
61%
61%
1-3
3-5
5-7
>7
Number of Goods
Based on the PPI data we group firms by the number of goods. We compute the mean fraction of positive price
changes in these groups in the following way. First, we compute the number of strictly positive good level price
changes over all zero and non-zero price changes for a given firm. Then, we report the mean across firms in a
given group. We compute upper and lower bounds of the error bands as ± 1.96 * std. error across firms.
33
Figure 4: Mean Absolute Size of Price Changes with 95% Bands
Mean Absolute Size of Price Changes with 95% Bands
9.00%
Absolute size of price changes
8.50%
8.00%
7.50%
7.00%
6.50%
6.00%
1-3
3-5
5-7
>7
Number of goods
Based on the PPI data we group firms by the number of goods. We compute the mean absolute size of price
changes in these groups in the following way. First, we compute the percentage change to last observed price
at the good level. Then, we compute the median size of price changes across goods at the firm level. Then, we
report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96
* std. error across firms.
34
Figure 5: Mean Size of Positive and Negative Price Changes with 95% Bands
Based on the PPI data we group firms by the number of goods. We compute the mean size of positive price
changes in these groups in the following way. First, we compute the percentage change to last observed price
at the good level. Then, we compute the median size of price changes across goods at the firm level. Then, we
report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96
* std. error across firms.
35
Figure 6: Mean Fraction of Small Price Changes with 95% Bands
Mean Fraction of Small Price Changes with 95% Bands
60%
Fraction of small price changes
55%
50%
45%
40%
35%
30%
1-3
3-5
5-7
>7
Number of goods
Based on the PPI data we group firms by the number of goods. We compute the mean fraction of small price
changes in these groups in the following way. First, we compute the fraction of price changes that are smaller
than 0.5 times the mean absolute percentage size of price changes across all goods in a firm. Then, we report
the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96 * std.
error across firms.
36
Figure 7: Mean Kurtosis of Price Changes with 95% Bands
Mean Kurtosis of Price Changes with 95% Bands
20
Kurtosis of price changes
18
16
14
12
10
8
6
4
1-3
3-5
5-7
>7
Number of goods
Based on the PPI data we group firms by the number of goods. We compute the mean kurtosis of price changes
in these groups in the following way. First, we compute the kurtosis of price changes at the firm level, defined as
the ratio of the fourth moment about the mean and the variance squared of percentage price changes. Then, we
report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96
* std. error across firms.
37
Figure 8: Mean First and 99th Percentile of Price Changes with 95% Bands
Based on the PPI data we group firms by the number of goods. We compute the mean size of positive price
changes in these groups in the following way. First, we compute the percentage change to last observed price
at the good level. Then, we compute the median size of price changes across goods at the firm level. Then, we
report the mean across firms in a given group. We compute upper and lower bounds of the error bands as ± 1.96
* std. error across firms.
38
APPENDIX A
This appendix contains empirical robustness results to which we make reference in the text.
Table A.1: Distribution of Firms across Sectors and Bins
Sector
11
21
22
23
31
32
33
42
44
45
48
49
51
52
53
54
56
62
71
Bin 1
58.1%
1.36%
74.48%
7.17%
19.11%
0.82%
48.25%
3.08%
28.01%
12%
34.17%
18.82%
28.72%
28.76%
27.23%
1.74%
35.28%
3.78%
28.12%
1.01%
36.74%
2.69%
45.03%
0.86%
30.79%
3.65%
28.07%
4.03%
59.56%
4.21%
37.34%
2.01%
29.81%
1.39%
21.06%
1.82%
26.87%
Bin 2
30.95%
0.5%
19.84%
1.32%
9.95%
0.29%
51.75%
2.29%
49.92%
14.79%
47.73%
18.17%
52.85%
36.59%
61.95%
2.74%
47.18%
3.49%
50.94%
1.26%
48.17%
2.44%
38.6%
0.51%
48.31%
3.96%
31.65%
3.14%
33.81%
1.65%
53.32%
1.99%
35.82%
1.15%
38.89%
2.33%
62.69%
Bin 3
Bin 4
Total
5.24%
5.71%
100%
0.32%
0.58%
0.77%
3.6%
2.09%
100%
0.91%
0.87%
3.15%
10.73% 60.21% 100%
1.21% 11.16%
1.4%
0%
0%
100%
0%
0%
2.09%
15.05% 7.02%
100%
16.96% 13.05% 14.02%
13.33% 4.77%
100%
19.31% 11.4% 18.01%
13.32% 5.11%
100%
35.1% 22.22% 32.76%
7.33%
3.49%
100%
1.23%
0.97%
2.09%
7.93%
9.6%
100%
2.23%
4.46%
3.5%
7.5%
13.44% 100%
0.71%
2.09%
1.17%
6.55%
8.54%
100%
1.26%
2.72%
2.4%
8.19%
8.19%
100%
0.41%
0.68%
0.63%
8.85% 12.05% 100%
2.76%
6.21%
3.88%
20.45% 19.83% 100%
7.73% 12.37%
4.7%
2.37%
4.27%
100%
0.44%
1.31%
2.31%
8.51%
0.83%
100%
1.21%
0.19%
1.76%
29.09% 5.29%
100%
3.56%
1.07%
1.52%
17.44% 22.61% 100%
3.97%
8.49%
2.83%
10.45%
0%
100%
Continued on next page
1
Table A.1 – continued from previous page
Sector Bin 1
Bin 2
Bin 3
Bin 4
Total
0.2%
0.32%
0.21%
0%
0.24%
72
25%
65.87% 7.69%
1.44%
100%
0.58%
1.06%
0.47%
0.15%
0.76%
Total
32.71% 47.32% 12.44% 7.53%
100%
100%
100%
100%
100%
100%
The table shows the percentage of firms in the PPI that belong to a bin in a given
two-digit NAICS category (each first line per industry) and the percentage in an
industry given a bin (each second line). Bin 1 groups firms with 1 to 3 goods, bin 2
firms with 3 to 5 goods, bin 3 firms with 5 to 7 goods and bin 4 firms with more than
7 goods
Table A.2: Distribution of Firms across Sectors and Bins
Sector
111
112
113
114
211
212
213
221
236
311
312
313
314
Bin 1
72.73%
0.72%
30%
0.07%
50%
0.49%
57.14%
0.09%
25.3%
0.23%
81.33%
5.89%
71.76%
1.05%
19.11%
0.82%
48.25%
3.08%
24.47%
5.43%
31.62%
1.03%
27.15%
1.13%
25.53%
0.95%
Bin 2
19.32%
0.13%
35%
0.05%
45.45%
0.31%
7.14%
0.01%
45.78%
0.29%
15.59%
0.78%
24.43%
0.25%
9.95%
0.29%
51.75%
2.29%
46.58%
7.15%
38.14%
0.86%
62.37%
1.79%
65.77%
1.69%
Bin 3
Bin 4
Total
3.41%
4.55% 100%
0.09%
0.19% 0.32%
15%
20%
100%
0.09%
0.19% 0.07%
4.55%
0%
100%
0.12%
0%
0.32%
7.14% 28.57% 100%
0.03%
0.19% 0.05%
15.66% 13.25% 100%
0.38%
0.53%
0.3%
2.16%
0.93% 100%
0.41%
0.29% 2.37%
3.05%
0.76% 100%
0.12%
0.05% 0.48%
10.73% 60.21% 100%
1.21% 11.16% 1.4%
0%
0%
100%
0%
0%
2.09%
18.93% 10.02% 100%
11.05% 9.66% 7.26%
21.65% 8.59% 100%
1.85%
1.21% 1.06%
8.06%
2.42% 100%
0.88%
0.44% 1.36%
5.71%
3%
100%
0.56%
0.49% 1.22%
Continued on next page
2
Table A.2 –
Sector Bin 1
315
34.63%
2.38%
316
40.93%
1.08%
321
36.64%
2.73%
322
29.45%
1.81%
323
24.08%
1.32%
324
63.48%
1.63%
325
29.09%
3.75%
326
24.11%
1.74%
327
43.87%
5.83%
331
35.74%
3.77%
332
32.08%
6.21%
333
24.21%
5.25%
334
29.14%
3.18%
335
27.93%
2.35%
30.99%
336
3.45%
337
21.9%
1.83%
339
27.96%
2.72%
421
36.97%
0.49%
423
24.9%
0.69%
424
25.29%
0.48%
20%
425
0.08%
441
41.24%
continued from previous page
Bin 2
Bin 3
Bin 4
Total
52.03% 9.92%
3.41% 100%
2.47%
1.79%
1.02% 2.25%
45.15% 11.81% 2.11% 100%
0.83%
0.82%
0.24% 0.87%
53.75% 7.21%
2.4%
100%
2.77%
1.41%
0.78% 2.43%
57.45%
10%
3.09% 100%
2.44%
1.62%
0.82% 2.01%
47.35% 27.55% 1.02% 100%
1.79%
3.97%
0.24% 1.79%
30%
0.87%
5.65% 100%
0.53%
0.06%
0.63% 0.84%
41.82% 20.78% 8.31% 100%
3.73%
7.05%
4.66% 4.22%
54.1% 11.75% 10.05% 100%
2.7%
2.23%
3.15% 2.37%
45.71% 8.49%
1.93% 100%
4.2%
2.97%
1.12% 4.35%
46.55% 13.79% 3.92% 100%
3.39%
3.82%
1.8%
3.45%
54.82% 9.69%
3.4%
100%
7.34%
4.94%
2.86% 6.33%
52.86% 17.26% 5.67% 100%
7.93%
9.85%
5.34%
7.1%
52.04% 12.88% 5.93% 100%
3.93%
3.7%
2.81% 3.58%
50.27% 13.56% 8.24% 100%
2.92%
3%
3.01% 2.75%
51.15% 13.24% 4.61% 100%
3.94%
3.88%
2.23% 3.64%
57.14% 16.82% 4.14% 100%
3.31%
3.7%
1.5%
2.74%
57.08% 8.63%
6.33% 100%
3.83%
2.2%
2.67% 3.18%
54.62% 5.04%
3.36% 100%
0.5%
0.18%
0.19% 0.44%
64.66% 8.03%
2.41% 100%
1.24%
0.59%
0.29% 0.91%
60%
8.82%
5.88% 100%
0.79%
0.44%
0.49% 0.62%
77.14% 2.86%
0%
100%
0.21%
0.03%
0%
0.13%
41.24% 5.15% 12.37% 100%
Continued on next page
3
Table A.2 –
Sector Bin 1
0.45%
442
79.07%
0.76%
443
67.39%
0.69%
27.03%
444
0.45%
445
13.41%
0.39%
446
7.25%
0.06%
41.67%
447
0.39%
43.8%
448
0.59%
451
33.33%
0.25%
452
50.7%
0.4%
453
14.49%
0.11%
454
19.3%
0.25%
481
44.44%
0.36%
482
15.15%
0.06%
483
47.14%
0.37%
484
28.39%
0.75%
486
12.77%
0.07%
488
49.49%
1.1%
491
0%
0%
492
21.67%
0.15%
493
58.18%
0.72%
511
31.49%
1.78%
continued from previous page
Bin 2
Bin 3
Bin 4
Total
0.31%
0.15%
0.58% 0.35%
19.77% 1.16%
0%
100%
0.13%
0.03%
0%
0.31%
28.26% 3.26%
1.09% 100%
0.2%
0.09%
0.05% 0.34%
67.57% 4.73%
0.68% 100%
0.77%
0.21%
0.05% 0.54%
60.15% 12.64% 13.79% 100%
1.21%
0.97%
1.75% 0.95%
62.32%
5.8%
24.64% 100%
0.33%
0.12%
0.82% 0.25%
45.24% 3.57%
9.52% 100%
0.29%
0.09%
0.39% 0.31%
25.62% 16.53% 14.05% 100%
0.24%
0.59%
0.82% 0.44%
34.85% 7.58% 24.24% 100%
0.18%
0.15%
0.78% 0.24%
22.54% 7.04% 19.72% 100%
0.12%
0.15%
0.68% 0.26%
65.22%
5.8%
14.49% 100%
0.35%
0.12%
0.49% 0.25%
69.3%
8.77%
2.63% 100%
0.61%
0.29%
0.15% 0.42%
20.83% 9.72%
25%
100%
0.12%
0.21%
0.87% 0.26%
21.21% 27.27% 36.36% 100%
0.05%
0.26%
0.58% 0.12%
32.86% 12.86% 7.14% 100%
0.18%
0.26%
0.24% 0.26%
61.44% 3.39%
6.78% 100%
1.12%
0.24%
0.78% 0.86%
82.98% 4.26%
0%
100%
0.3%
0.06%
0%
0.17%
43.94% 4.04%
2.53% 100%
0.67%
0.24%
0.24% 0.72%
0%
0%
100%
100%
0%
0%
0.05%
0%
51.67%
5%
21.67% 100%
0.24%
0.09%
0.63% 0.22%
31.82%
10%
0%
100%
0.27%
0.32%
0%
0.4%
55.25% 6.14%
7.13% 100%
2.16%
0.91%
1.75% 1.85%
Continued on next page
4
Table A.2 –
Sector Bin 1
515
30.05%
0.61%
517
14.48%
0.36%
518
52.94%
0.91%
522
54.48%
1.63%
523
42.86%
0.94%
524
15.94%
1.46%
531
71.55%
3.82%
532
22.58%
0.39%
541
37.34%
2.01%
561
29.8%
1.35%
562
30%
0.03%
621
22.49%
0.73%
622
7.21%
0.26%
623
45.18%
0.84%
26.87%
713
0.2%
721
25%
0.58%
Total
32.71%
100%
continued from previous page
Bin 2
Bin 3
Bin 4
Total
62.3%
7.65%
0%
100%
0.88%
0.41%
0%
0.67%
23.53% 20.36% 41.63% 100%
0.4%
1.32%
4.46% 0.81%
44.44% 2.61%
0%
100%
0.53%
0.12%
0%
0.56%
36.94% 6.34%
2.24% 100%
0.76%
0.5%
0.29% 0.98%
43.88% 4.08%
9.18% 100%
0.66%
0.24%
0.87% 0.72%
27.01% 28.95% 28.1% 100%
1.72%
7%
11.21%
3%
25.1%
1.67%
1.67% 100%
0.93%
0.24%
0.39% 1.75%
60.65% 4.52% 12.26% 100%
0.73%
0.21%
0.92% 0.57%
53.32% 8.51%
0.83% 100%
1.99%
1.21%
0.19% 1.76%
34.98% 29.8%
5.42% 100%
1.1%
3.56%
1.07% 1.48%
70%
0%
0%
100%
0.05%
0%
0%
0.04%
60.55% 8.65%
8.3%
100%
1.35%
0.73%
1.16% 1.06%
11.91% 34.48% 46.39% 100%
0.29%
3.23%
7.18% 1.17%
53.01%
0%
1.81% 100%
0.68%
0%
0.15% 0.61%
62.69% 10.45%
0%
100%
0.32%
0.21%
0%
0.24%
65.87% 7.69%
1.44% 100%
1.06%
0.47%
0.15% 0.76%
47.32% 12.44% 7.53% 100%
100%
100%
100%
100%
The table shows the percentage of firms in the PPI that belong to a bin in a given
three-digit NAICS category (each first line per industry) and the percentage in an
industry given a bin (each second line). Bin 1 groups firms with 1 to 3 goods, bin 2
firms with 3 to 5 goods, bin 3 firms with 5 to 7 goods and bin 4 firms with more than
7 goods.
5
Table A.3: Distribution of Firms across Sectors and Bins
Sector
1111
1112
1113
1119
1121
1122
1123
1124
1133
1141
2111
2121
2122
2123
2131
2211
2212
2362
3111
3112
3113
Bin 1
66.67%
0.16%
70.97%
0.25%
77.42%
0.27%
80%
0.04%
60%
0.03%
50%
0.01%
18.18%
0.02%
0%
0%
50%
0.49%
57.14%
0.09%
25.3%
0.23%
74.19%
1.03%
93.94%
0.35%
82.28%
4.51%
71.76%
1.05%
22.56%
0.67%
11.21%
0.15%
48.25%
3.08%
20.57%
0.48%
23.03%
0.46%
45.76%
Bin 2
19.05%
0.03%
22.58%
0.05%
16.13%
0.04%
20%
0.01%
0%
0%
0%
0%
54.55%
0.05%
50%
0.01%
45.45%
0.31%
7.14%
0.01%
45.78%
0.29%
21.77%
0.21%
6.06%
0.02%
14.66%
0.56%
24.43%
0.25%
12.78%
0.26%
3.45%
0.03%
51.75%
2.29%
49.28%
0.8%
44.94%
0.62%
19.49%
Bin 3
Bin 4
Total
9.52%
4.76% 100%
0.06%
0.05% 0.08%
0%
6.45% 100%
0%
0.1%
0.11%
3.23%
3.23% 100%
0.03%
0.05% 0.11%
0%
0%
100%
0%
0%
0.02%
0%
40%
100%
0%
0.1%
0.02%
50%
0%
100%
0.03%
0%
0.01%
9.09% 18.18% 100%
0.03%
0.1%
0.04%
50%
0%
100%
0.03%
0%
0.01%
4.55%
0%
100%
0.12%
0%
0.32%
7.14% 28.57% 100%
0.03%
0.19% 0.05%
15.66% 13.25% 100%
0.38%
0.53%
0.3%
4.03%
0%
100%
0.15%
0%
0.45%
0%
0%
100%
0%
0%
0.12%
1.83%
1.22% 100%
0.26%
0.29% 1.79%
3.05%
0.76% 100%
0.12%
0.05% 0.48%
12.41% 52.26% 100%
0.97%
6.74% 0.97%
6.9%
78.45% 100%
0.24%
4.42% 0.42%
0%
0%
100%
0%
0%
2.09%
16.75% 13.4% 100%
1.03%
1.36% 0.76%
19.1% 12.92% 100%
1%
1.12% 0.65%
27.97% 6.78% 100%
Continued on next page
6
Table A.3 – continued from previous page
Sector Bin 1
Bin 2
Bin 3
Bin 4
Total
0.6%
0.18%
0.97%
0.39% 0.43%
3114
29.15% 46.86% 13.65% 10.33% 100%
0.88%
0.98%
1.09%
1.36% 0.99%
3115
16.12% 38.79% 31.23% 13.85% 100%
0.72%
1.19%
3.64%
2.67% 1.45%
19.72% 55.63% 18.66% 5.99% 100%
3116
0.63%
1.22%
1.56%
0.82% 1.04%
3117
65.71%
20%
10.48% 3.81% 100%
0.77%
0.16%
0.32%
0.19% 0.38%
3118
17.73% 54.09% 14.09% 14.09% 100%
0.44%
0.92%
0.91%
1.5%
0.8%
20.1% 68.63% 8.82%
2.45% 100%
3119
0.46%
1.08%
0.53%
0.24% 0.75%
27.32% 40.49% 22.44% 9.76% 100%
3121
0.63%
0.64%
1.35%
0.97% 0.75%
3122
41.86% 32.56% 19.77% 5.81% 100%
0.4%
0.22%
0.5%
0.24% 0.31%
3131
29.23% 56.92% 10.77% 3.08% 100%
0.21%
0.29%
0.21%
0.1%
0.24%
3132
31.22% 59.92% 6.75%
2.11% 100%
0.83%
1.1%
0.47%
0.24% 0.87%
3133
11.43% 75.71%
10%
2.86% 100%
0.09%
0.41%
0.21%
0.1%
0.26%
3141
23.33% 61.33% 9.33%
6%
100%
0.39%
0.71%
0.41%
0.44% 0.55%
3149
27.32% 69.4%
2.73%
0.55% 100%
0.56%
0.98%
0.15%
0.05% 0.67%
3151
17.7% 53.98% 23.01% 5.31% 100%
0.22%
0.47%
0.76%
0.29% 0.41%
3152
44.3% 45.57% 7.09%
3.04% 100%
1.96%
1.39%
0.82%
0.58% 1.44%
3159
16.82% 73.83% 6.54%
2.8%
100%
0.2%
0.61%
0.21%
0.15% 0.39%
3161
30.88% 57.35% 4.41%
7.35% 100%
0.23%
0.3%
0.09%
0.24% 0.25%
3162
27.78% 44.44% 27.78%
0%
100%
0.17%
0.19%
0.44%
0%
0.2%
3169
53.04% 38.26%
8.7%
0%
100%
0.68%
0.34%
0.29%
0%
0.42%
3211
13.41% 67.07% 13.41%
6.1%
100%
0.25%
0.85%
0.65%
0.49%
0.6%
3212
61.61% 28.91% 6.64%
2.84% 100%
1.45%
0.47%
0.41%
0.29% 0.77%
Continued on next page
7
Table A.3 – continued from previous page
Sector Bin 1
Bin 2
Bin 3
Bin 4
Total
3219
31.62% 64.26% 4.12%
0%
100%
1.03%
1.44%
0.35%
0%
1.06%
3221
37.5% 28.12%
25%
9.38% 100%
0.27%
0.14%
0.47%
0.29% 0.23%
3222
28.4% 61.32% 8.02%
2.26% 100%
1.54%
2.3%
1.15%
0.53% 1.78%
3231
24.08% 47.35% 27.55% 1.02% 100%
1.32%
1.79%
3.97%
0.24% 1.79%
3241
63.48%
30%
0.87%
5.65% 100%
1.63%
0.53%
0.06%
0.63% 0.84%
3251
35.05% 45.33% 16.36% 3.27% 100%
0.84%
0.75%
1.03%
0.34% 0.78%
3252
33.08% 34.59% 22.56% 9.77% 100%
0.49%
0.36%
0.88%
0.63% 0.49%
3253
31.4% 53.49% 12.79% 2.33% 100%
0.3%
0.36%
0.32%
0.1%
0.31%
3254
38.12% 25.25% 18.32% 18.32% 100%
0.86%
0.39%
1.09%
1.8%
0.74%
3255
16.37% 51.46% 25.15% 7.02% 100%
0.31%
0.68%
1.26%
0.58% 0.63%
3256
17.9% 31.48% 40.74% 9.88% 100%
0.32%
0.39%
1.94%
0.78% 0.59%
3259
29.95% 55.61% 9.63%
4.81% 100%
0.63%
0.8%
0.53%
0.44% 0.68%
3261
27.2%
60.2%
7.05%
5.54% 100%
1.21%
1.85%
0.82%
1.07% 1.45%
3262
19.2%
44.4%
19.2%
17.2% 100%
0.54%
0.86%
1.41%
2.09% 0.91%
29.59% 57.53% 10.41% 2.47% 100%
3271
1.21%
1.62%
1.12%
0.44% 1.33%
3272
34.36% 52.86% 10.13% 2.64% 100%
0.87%
0.93%
0.68%
0.29% 0.83%
3273
56.84%
37%
4.83%
1.34% 100%
2.37%
1.07%
0.53%
0.24% 1.36%
3274
38%
50%
12%
0%
100%
0.21%
0.19%
0.18%
0%
0.18%
3279
60%
29.14% 9.14%
1.71% 100%
1.17%
0.39%
0.47%
0.15% 0.64%
3311
25.74% 23.76% 36.63% 13.86% 100%
0.29%
0.19%
1.09%
0.68% 0.37%
30.23% 49.42% 18.6%
1.74% 100%
3312
0.58%
0.66%
0.94%
0.15% 0.63%
3313
41.27% 42.06% 9.52%
7.14% 100%
Continued on next page
8
Table A.3 – continued from previous page
Sector Bin 1
Bin 2
Bin 3
Bin 4
Total
0.58%
0.41%
0.35%
0.44% 0.46%
3314
49.74% 39.49% 8.21%
2.56% 100%
1.08%
0.59%
0.47%
0.24% 0.71%
3315
31.52% 57.31% 9.46%
1.72% 100%
1.23%
1.55%
0.97%
0.29% 1.28%
50.79% 46.03% 1.59%
1.59% 100%
3321
0.72%
0.45%
0.06%
0.1%
0.46%
3322
23.85% 43.85% 31.54% 0.77% 100%
0.35%
0.44%
1.21%
0.05% 0.48%
3323
34.84% 60.37% 3.19%
1.6%
100%
1.46%
1.75%
0.35%
0.29% 1.37%
30.95% 49.21% 13.49% 6.35% 100%
3324
0.44%
0.48%
0.5%
0.39% 0.46%
25%
47.5%
25%
2.5%
100%
3325
0.22%
0.29%
0.59%
0.1%
0.29%
3326
32.17% 66.09% 1.74%
0%
100%
0.41%
0.59%
0.06%
0%
0.42%
3327
25.95% 70.89% 3.16%
0%
100%
0.46%
0.87%
0.15%
0%
0.58%
3328
36.07% 61.2%
2.73%
0%
100%
0.74%
0.87%
0.15%
0%
0.67%
3329
28.93% 47.38% 14.58% 9.11% 100%
1.42%
1.61%
1.88%
1.94%
1.6%
3331
15.51% 46.12% 22.86% 15.51% 100%
0.42%
0.87%
1.65%
1.84%
0.9%
3332
33.82% 53.06% 11.08% 2.04% 100%
1.3%
1.41%
1.12%
0.34% 1.25%
3333
23.74% 57.55% 15.83% 2.88% 100%
0.37%
0.62%
0.65%
0.19% 0.51%
3334
20.32% 60.43% 18.18% 1.07% 100%
0.42%
0.87%
1%
0.1%
0.68%
3335
29.89% 62.36% 5.54%
2.21% 100%
0.91%
1.31%
0.44%
0.29% 0.99%
3336
22.16% 52.1% 18.56% 7.19% 100%
0.41%
0.67%
0.91%
0.58% 0.61%
3339
21.56% 47.88% 23.6%
6.96% 100%
1.42%
2.18%
4.09%
1.99% 2.15%
3341
42.48% 37.17% 12.39% 7.96% 100%
0.54%
0.32%
0.41%
0.44% 0.41%
3342
22.73% 62.5% 11.36% 3.41% 100%
0.22%
0.42%
0.29%
0.15% 0.32%
3343
35%
55%
10%
0%
100%
0.08%
0.08%
0.06%
0%
0.07%
Continued on next page
9
Table A.3 – continued from previous page
Sector Bin 1
Bin 2
Bin 3
Bin 4
Total
3344
25.86% 52.02% 14.33% 7.79% 100%
0.93%
1.29%
1.35%
1.21% 1.17%
3345
28.35% 53.92% 13.16% 4.56% 100%
1.25%
1.65%
1.53%
0.87% 1.44%
3346
36.59% 51.22% 4.88%
7.32% 100%
0.17%
0.16%
0.06%
0.15% 0.15%
3351
46.28% 40.43% 11.17% 2.13% 100%
0.97%
0.59%
0.62%
0.19% 0.69%
3352
20.55% 45.21% 26.03% 8.22% 100%
0.17%
0.25%
0.56%
0.29% 0.27%
3353
24.37% 56.3% 10.92%
8.4%
100%
0.65%
1.04%
0.76%
0.97% 0.87%
3359
19.76% 53.36% 14.23% 12.65% 100%
0.56%
1.04%
1.06%
1.55% 0.92%
3361
25%
53.12% 9.38%
12.5% 100%
0.09%
0.13%
0.09%
0.19% 0.12%
3362
36.9%
50.8%
10.7%
1.6%
100%
0.77%
0.73%
0.59%
0.15% 0.68%
3363
16.32% 60.14% 17.48% 6.06% 100%
0.78%
1.99%
2.2%
1.26% 1.57%
3364
38.97%
50%
6.62%
4.41% 100%
0.59%
0.53%
0.26%
0.29%
0.5%
3365
24.32% 64.86% 10.81%
0%
100%
0.1%
0.19%
0.12%
0%
0.14%
3366
62.59% 25.9%
7.91%
3.6%
100%
0.97%
0.28%
0.32%
0.24% 0.51%
3369
35.14% 32.43% 27.03% 5.41% 100%
0.15%
0.09%
0.29%
0.1%
0.14%
22.66% 59.05% 15.31% 2.98% 100%
3371
1.27%
2.29%
2.26%
0.73% 1.84%
3372
20%
58.18% 12.12%
9.7%
100%
0.37%
0.74%
0.59%
0.78%
0.6%
3379
20.99% 43.21% 35.8%
0%
100%
0.19%
0.27%
0.85%
0%
0.3%
3391
22.53% 55.97% 11.6%
9.9%
100%
0.74%
1.27%
1%
1.41% 1.07%
3399
30.73% 57.64% 7.12%
4.51% 100%
1.98%
2.56%
1.21%
1.26% 2.11%
4219
36.97% 54.62% 5.04%
3.36% 100%
0.49%
0.5%
0.18%
0.19% 0.44%
24.9% 64.66% 8.03%
2.41% 100%
4230
0.69%
1.24%
0.59%
0.29% 0.91%
4240
25.29%
60%
8.82%
5.88% 100%
Continued on next page
10
Table A.3 – continued from previous page
Sector Bin 1
Bin 2
Bin 3
Bin 4
Total
0.48%
0.79%
0.44%
0.49% 0.62%
4251
20%
77.14% 2.86%
0%
100%
0.08%
0.21%
0.03%
0%
0.13%
4411
52.38% 4.76% 14.29% 28.57% 100%
0.12%
0.01%
0.09%
0.29% 0.08%
47.92% 47.92%
0%
4.17% 100%
4412
0.26%
0.18%
0%
0.1%
0.18%
4413
21.43% 57.14% 7.14% 14.29% 100%
0.07%
0.12%
0.06%
0.19%
0.1%
4421
84.75% 13.56% 1.69%
0%
100%
0.56%
0.06%
0.03%
0%
0.22%
66.67% 33.33%
0%
0%
100%
4422
0.2%
0.07%
0%
0%
0.1%
67.39% 28.26% 3.26%
1.09% 100%
4431
0.69%
0.2%
0.09%
0.05% 0.34%
4441
26.89% 68.07%
4.2%
0.84% 100%
0.36%
0.63%
0.15%
0.05% 0.44%
4442
27.59% 65.52%
6.9%
0%
100%
0.09%
0.15%
0.06%
0%
0.11%
4451
4.72% 44.34% 25.47% 25.47% 100%
0.06%
0.36%
0.79%
1.31% 0.39%
4452
21.21% 68.94% 4.55%
5.3%
100%
0.31%
0.7%
0.18%
0.34% 0.48%
4453
8.7%
82.61%
0%
8.7%
100%
0.02%
0.15%
0%
0.1%
0.08%
4461
7.25% 62.32%
5.8%
24.64% 100%
0.06%
0.33%
0.12%
0.82% 0.25%
4471
41.67% 45.24% 3.57%
9.52% 100%
0.39%
0.29%
0.09%
0.39% 0.31%
4481
55.07% 20.29% 18.84%
5.8%
100%
0.42%
0.11%
0.38%
0.19% 0.25%
4482
60%
15%
5%
20%
100%
0.13%
0.02%
0.03%
0.19% 0.07%
4483
9.38% 43.75% 18.75% 28.12% 100%
0.03%
0.11%
0.18%
0.44% 0.12%
4511
12.2%
43.9%
12.2% 31.71% 100%
0.06%
0.14%
0.15%
0.63% 0.15%
4512
68%
20%
0%
12%
100%
0.19%
0.04%
0%
0.15% 0.09%
4521
50%
25%
10%
15%
100%
0.22%
0.08%
0.12%
0.29% 0.15%
4529
51.61% 19.35% 3.23% 25.81% 100%
0.18%
0.05%
0.03%
0.39% 0.11%
Continued on next page
11
Table A.3 – continued from previous page
Sector Bin 1
Bin 2
Bin 3
Bin 4
Total
4531
11.11% 88.89%
0%
0%
100%
0.02%
0.12%
0%
0%
0.07%
4532
3.03% 54.55% 12.12% 30.3% 100%
0.01%
0.14%
0.12%
0.49% 0.12%
4539
38.89% 61.11%
0%
0%
100%
0.08%
0.08%
0%
0%
0.07%
4541
13.51% 75.68% 6.76%
4.05% 100%
0.11%
0.43%
0.15%
0.15% 0.27%
4542
16.67% 72.22% 11.11%
0%
100%
0.03%
0.1%
0.06%
0%
0.07%
4543
40.91% 45.45% 13.64%
0%
100%
0.1%
0.08%
0.09%
0%
0.08%
4811
16.28% 25.58% 16.28% 41.86% 100%
0.08%
0.08%
0.21%
0.87% 0.16%
4812
86.21% 13.79%
0%
0%
100%
0.28%
0.03%
0%
0%
0.11%
4821
15.15% 21.21% 27.27% 36.36% 100%
0.06%
0.05%
0.26%
0.58% 0.12%
4831
31.43% 42.86% 11.43% 14.29% 100%
0.12%
0.12%
0.12%
0.24% 0.13%
4832
62.86% 22.86% 14.29%
0%
100%
0.25%
0.06%
0.15%
0%
0.13%
4841
20.83% 62.5%
5%
11.67% 100%
0.28%
0.58%
0.18%
0.68% 0.44%
4842
36.21% 60.34% 1.72%
1.72% 100%
0.47%
0.54%
0.06%
0.1%
0.42%
4861
28.57% 66.67% 4.76%
0%
100%
0.07%
0.11%
0.03%
0%
0.08%
0%
96.15% 3.85%
0%
100%
4869
0%
0.19%
0.03%
0%
0.1%
4881
53.33% 28.33%
10%
8.33% 100%
0.36%
0.13%
0.18%
0.24% 0.22%
4883
48.15%
50%
1.85%
0%
100%
0.58%
0.42%
0.06%
0%
0.39%
4885
46.67% 53.33%
0%
0%
100%
0.16%
0.12%
0%
0%
0.11%
4911
0%
0%
0%
100%
100%
0%
0%
0%
0.05%
0%
4921
21.21% 33.33% 9.09% 36.36% 100%
0.08%
0.08%
0.09%
0.58% 0.12%
22.22% 74.07%
0%
3.7%
100%
4922
0.07%
0.15%
0%
0.05%
0.1%
4931
58.18% 31.82%
10%
0%
100%
Continued on next page
12
Table A.3 – continued from previous page
Sector Bin 1
Bin 2
Bin 3
Bin 4
Total
0.72%
0.27%
0.32%
0%
0.4%
5111
24.01% 60.64% 6.93%
8.42% 100%
1.08%
1.89%
0.82%
1.65% 1.48%
5112
61.39% 33.66% 2.97%
1.98% 100%
0.69%
0.26%
0.09%
0.1%
0.37%
30.06% 62.58% 7.36%
0%
100%
5151
0.55%
0.79%
0.35%
0%
0.6%
5152
30%
60%
10%
0%
100%
0.07%
0.09%
0.06%
0%
0.07%
5171
0%
3.37% 13.48% 83.15% 100%
0%
0.02%
0.35%
3.59% 0.33%
55%
5%
10%
30%
100%
5172
0.12%
0.01%
0.06%
0.29% 0.07%
18.75% 42.86% 27.68% 10.71% 100%
5175
0.23%
0.37%
0.91%
0.58% 0.41%
5181
52.44% 42.68% 4.88%
0%
100%
0.48%
0.27%
0.12%
0%
0.3%
5182
53.52% 46.48%
0%
0%
100%
0.42%
0.25%
0%
0%
0.26%
5221
54.48% 36.94% 6.34%
2.24% 100%
1.63%
0.76%
0.5%
0.29% 0.98%
5231
45.71% 40.95% 4.76%
8.57% 100%
0.54%
0.33%
0.15%
0.44% 0.38%
5239
39.56% 47.25%
3.3%
9.89% 100%
0.4%
0.33%
0.09%
0.44% 0.33%
5241
13.77% 21.81% 31.56% 32.86% 100%
1.07%
1.17%
6.47% 11.11% 2.55%
5242
28%
56%
14.4%
1.6%
100%
0.39%
0.54%
0.53%
0.1%
0.46%
5311
94.6%
5.04%
0%
0.36% 100%
2.94%
0.11%
0%
0.05% 1.02%
5312
38.24% 51.96% 5.88%
3.92% 100%
0.44%
0.41%
0.18%
0.19% 0.37%
5313
40.82% 54.08% 2.04%
3.06% 100%
0.45%
0.41%
0.06%
0.15% 0.36%
5321
28.95% 43.42% 2.63%
25%
100%
0.25%
0.25%
0.06%
0.92% 0.28%
5324
16.46% 77.22% 6.33%
0%
100%
0.15%
0.47%
0.15%
0%
0.29%
5411
28.43% 59.31% 10.29% 1.96% 100%
0.65%
0.93%
0.62%
0.19% 0.75%
5412
48.67% 46.02% 5.31%
0%
100%
0.61%
0.4%
0.18%
0%
0.41%
Continued on next page
13
Table A.3 – continued from previous page
Sector Bin 1
Bin 2
Bin 3
Bin 4
Total
5413
52.63% 33.33% 14.04%
0%
100%
0.34%
0.15%
0.24%
0%
0.21%
5416
33.87% 61.29% 4.84%
0%
100%
0.23%
0.29%
0.09%
0%
0.23%
5418
34.78% 58.7%
6.52%
0%
100%
0.18%
0.21%
0.09%
0%
0.17%
5613
23.38% 44.81% 24.68% 7.14% 100%
0.4%
0.53%
1.12%
0.53% 0.56%
5615
24.07% 25.93% 48.15% 1.85% 100%
0.15%
0.11%
0.76%
0.05%
0.2%
5616
35.53% 18.42% 42.11% 3.95% 100%
0.3%
0.11%
0.94%
0.15% 0.28%
5617
36.89% 36.89% 20.49% 5.74% 100%
0.5%
0.35%
0.73%
0.34% 0.45%
5621
30%
70%
0%
0%
100%
0.03%
0.05%
0%
0%
0.04%
6211
20.13% 54.55% 13.64% 11.69% 100%
0.35%
0.65%
0.62%
0.87% 0.56%
6216
25.19% 67.41% 2.96%
4.44% 100%
0.38%
0.7%
0.12%
0.29% 0.49%
6221
7.38% 11.07% 27.31% 54.24% 100%
0.22%
0.23%
2.18%
7.13% 0.99%
6222
7.32% 19.51% 73.17%
0%
100%
0.03%
0.06%
0.88%
0%
0.15%
6223
0%
0%
85.71% 14.29% 100%
0%
0%
0.18%
0.05% 0.03%
6231
43.31% 54.78%
0%
1.91% 100%
0.76%
0.66%
0%
0.15% 0.57%
77.78% 22.22%
0%
0%
100%
6232
0.08%
0.02%
0%
0%
0.03%
7131
0%
88.89% 11.11%
0%
100%
0%
0.06%
0.03%
0%
0.03%
7139
31.03% 58.62% 10.34%
0%
100%
0.2%
0.26%
0.18%
0%
0.21%
7211
25%
65.87% 7.69%
1.44% 100%
0.58%
1.06%
0.47%
0.15% 0.76%
Total
32.71% 47.32% 12.44% 7.53% 100%
100%
100%
100%
100%
100%
The table shows the percentage of firms in the PPI that belong to a bin in a given
four-digit NAICS category (each first line per industry) and the percentage in an
industry given a bin (each second line). Bin 1 groups firms with 1 to 3 goods, bin 2
firms with 3 to 5 goods, bin 3 firms with 5 to 7 goods and bin 4 firms with more than
7 goods.
14
Table A.4: Variation Explained by Fixed Effects
R2
Adjusted R2
Month FEs
Product FEs
Firm FEs
Products at four-digit level
3.20% 10.39%
26.95%
3.20% 10.39%
25.31%
Yes
Yes
Yes
No
Yes
Yes
No
No
Yes
Products at six-digit level
3.20% 13.56% 27.64%
3.20% 13.47% 25.80%
Yes
Yes
Yes
No
Yes
Yes
No
No
Yes
Products at eight-digit level
3.20% 15.68%
28.54%
3.20% 15.48%
26.41%
Yes
Yes
Yes
No
Yes
Yes
No
No
Yes
We estimate the following specification regarding variation in |∆p|, the absolute size of price changes: |∆p|i,f,p,t =
P roduct
F irm
α0 {DM onth m }m=12
+ α2 Di,f,p
+ i,f,p,t where i denotes a good, f a firm, p a product at the 4-, 6-, and 8-digit level,
m=1 + α1 Di,f,p,t
and t time. Dummies are for months, products, and firms.
Table A.5: Relation of Frequency and Number of Goods, Six-Digit Level
Mean
Median
25% Percentile
75% Percentile
Estimated Coefficient
1.22
(0.12)
1.32
0.59
1.80
We estimate the following specification: fj,p =
β0,p + β1,p nj,p + j,p , where fj,p denotes the frequency of firm j in a six-digit product category
p and nj,p the number of goods of that firm. We
estimate the specification at each date, take the
median of β1,p across all products at that date and
report the mean, median and quartiles of median
β1,p over time.
15
Table A.6: Relation of Absolute Size of Price Changes and Number of Goods, Six-Digit Level
Mean
Median
25% Percentile
75% Percentile
Estimated Coefficient
-0.38
(0.026)
-0.358
-0.521
-0.0315
We estimate the following specification: |∆p|j,p =
β0,p + β1,p nj,p + j,p , where |∆p|j,p denotes the
mean absolute size of price changes of firm j in a
six-digit product category p and nj,p the number
of goods of that firm. We estimate the specification at each date, take the mean of β1,p across
all products at that date and report the mean,
median and quartiles of median β1,p over time.
Table A.7: Fraction of Small Price Changes According to Different Definitions
Number
of Goods
1-3
3-5
5-7
>7
Pooled
|dp| < 12 |dp|
39.46%
(0.32%)
44.61%
(0.31%)
47.45%
(0.54%)
50.22%
(0.61%)
42.93%
(0.2%)
|dp| < 13 |dp|
31.70%
(0.32%)
36.52%
(0.32%)
39.20%
(0.58%)
42.41%
(0.69%)
34.98%
(0.2%)
|dp| < 14 |dp|
27.28%
(0.31%)
31.74%
(0.32%)
34.80%
(0.59%)
37.56%
(0.73%)
30.40%
(0.2%)
1
|dp| < 10
|dp|
17.95%
(0.29%)
21.53%
(0.31%)
23.69%
(0.58%)
26.41%
(0.76%)
20.44%
(0.19%)
|dp| < 1%
32.74%
(0.39%)
35.98%
(0.4%)
37.53%
(0.73%)
40.97%
(0.94%)
34.98%
(0.25%)
|dp| < 0.5%
25.78%
(0.37%)
28.81%
(0.38%)
30.15%
(0.71%)
33.40%
(0.94%)
27.86%
(0.24%)
|dp| < 0.25%
20.19%
(0.34%)
22.86%
(0.36%)
23.75%
(0.66%)
26.77%
(0.89%)
21.98%
(0.23%)
The fraction of small price changes is computed in the following way: First, we compute a cut-off that defines a small price change.
This is either a fraction of the mean absolute size of price changes |dp| for a given firm, as indicated in the columns, or an absolute
percentage number. Second, we compute the fraction of absolute price changes falling below the cut-off. Third, we summarize
means across firms within each bin.
16
Table A.8: Marginal Effects for Two-Digit Industries, ± 1/2 Std. Dev., Multinomial Logit
Pooled
1-3 Goods
3-5 Goods
5-7 Goods
>7 Goods
-0.30%
9.83%
1.11%
8.13%
0.68%
8.02%
-0.58%
11.53%
-3.59%
16.22%
0.01%
15.84%
0.97%
13.57%
0.25%
13.61%
-1.10%
17.79%
-3.09%
25.73%
Negative Changes
Fraction Industry
Fraction Firm
Positive Changes
Fraction Industry
Fraction Firm
The table shows the marginal effects in percentage points of a one-standard deviation change in the explanators
around the mean on the probability of adjusting prices upwards or downwards. Marginal effects are calculated for
the model described in the main text but for each bin separately and with a fraction of same-signed industry-level
price changes defined at the two-digit level. Estimation, given by the separate columns, is for the pooled data and
each bin separately. All reported effects are statistically significantly different from zero.
17
Figure A.1: Mean Frequency Size of Price Changes and Number of Goods, Controlling for Size
28.00%
Monthly Frequency of Price Changes and Number of Goods,
Controlling for Firm Size
Monthly Frequency of Price Changes
27.00%
26.00%
25.00%
24.00%
23.00%
22.00%
21.00%
20.00%
19.00%
1-3
3-5
Number of Goods
5-7
To obtain the frequency value shown, we estimate the following specification: fi = β0 employmenti +
>7
P
k
βk Dk,i + i where fi
is the median frequency of price adjustment for a firm i, employment the number of employees of the firm and Dk,i a dummy
for bin k that the firm is in. We then graph βk .
18
Figure A.2: Mean Absolute Size of Price Changes and Number of Goods, Controlling for Size
Mean Absolute Size of Price Changes
8.65%
Mean Absolute Size of Price Changes and Number of Goods,
Controlling for Size
8.15%
7.65%
7.15%
6.65%
6.15%
1-3
3-5
Number of Goods
5-7
>7
To obtain the mean absolute size of price change values shown, we estimate the following specification: |∆pi | = β0 employmenti +
P
k βk Dk,i + i where |∆pi | is the median absolute size of price changes for a firm i, employment the number of employees of
the firm and Dk,i a dummy for bin k that the firm is in. We then show βk .
19
Figure A.3: Increments in Mean Frequency of Price Changes Controlling for Sector Fixed Effects,
Relative to Baseline
Increment in Monthly Frequency Relative to Baseline
8.00%
Monthly Frequency of Price Changes and Number of Goods,
Controlling for Industry Characteristics
7.00%
6.00%
5.00%
4.00%
3.00%
2.00%
With two-digit industry controls
With three-digit industry controls
1.00%
With four-digit industry controls
0.00%
1-3
3-5
Number of Goods
5-7
To obtain the frequency values shown, we estimate the following specification: fi =
>7
P
k
βk Dk,i +
P
j
betaj IN Dj + i where fi
is the median frequency of price changes for a firm i, IN D a dummy variable for an industry defined at the 2-, 3-, or 4-digit
level. We then graph βk − min({β1 }).
20
Figure A.4: Increments in Mean Absolute Size of Price Changes Controlling for Sector Fixed Effects,
Relative to Baseline
Mean Absolute Relative Size of Price Changes
0.03
Mean Absolute Size of Price Changes and Number of Goods,
Relative to Baseline
0.025
0.02
0.015
0.01
With two-digit industry controls
0.005
With three-digit industry controls
With four-digit industry controls
0
1-3
3-5
Number of Goods
5-7
To obtain the frequency values shown, we estimate the following specification: |∆pi | =
>7
P
k
βk Dk,i +
P
j
betaj IN Dj + i where
|∆pi | is the median absolute size of price changes for a firm i, IN D a dummy variable for an industry defined at the 2-, 3-, or
4-digit level. We then graph βk − min({β4 }).
21
Figure A.5: Mean Frequency of Price Changes and Sectoral Decomposition
Monthly Frequency of Price Changes
Monthly Frequency of Price Changes
40.00%
30.00%
20.00%
Sectors 31, 32 ,33
Not sectors 31, 32, 33
10.00%
1-3
3-5
5-7
>7
Number of Goods
We compute the frequency of price changes in exactly the same way as for Figure 1 but with one change: in the last step of
aggregating firm frequencies, we take means for bin-sector group combinations as shown. The two relevant sector groupings are
the two-digit NAICS manufacturing sectors, and all others.
22
Figure A.6: Mean Frequency of Price Changes and Manufacturing Sectors 31, 32, 33
Monthly Frequency of Price Changes
Monthly Frequency of Price Changes
40.00%
30.00%
20.00%
Sector 31
Sector 32
Sector 33
10.00%
1-3
3-5
5-7
>7
Number of Goods
We compute the frequency of price changes in exactly the same way as for Figure 1 but with one change: in the last step of
aggregating firm frequencies, we take means for bin-sector combinations as shown. The relevant sectors are the two-digit NAICS
manufacturing sectors 31, 32 and 33.
23
Figure A.7: Mean Absolute Size of Price Changes and Manufacturing Sectors 31, 32, 33
Absolute Size of Price Changes
11.00%
Absolute Size of Price Changes
10.00%
9.00%
8.00%
7.00%
Sectors 31, 32 ,33
Not sectors 31, 32, 33
6.00%
5.00%
1-3
3-5
5-7
>7
Number of Goods
We compute the absolute size of price changes in exactly the same way as for Figure 4 but with one change: in the last step of
aggregating firm size of price change measures, we take means for bin-sector group combinations as shown. The two relevant
sector groupings are the two-digit NAICS manufacturing sectors, and all others.
24
Figure A.8: Mean Absolute Size of Price Changes and Manufacturing Sectors 31, 32, 33
Absolute Size of Price Changes
Absolute Size of Price Changes
7.50%
6.50%
5.50%
Sector 31
Sector 32
Sector 33
4.50%
1-3
3-5
5-7
>7
Number of Goods
We compute the absolute size of price changes in exactly the same way as for Figure 4 but with one change: in the last step of
aggregating firm size of price change measures, we take means for bin-sector combinations as shown. The relevant sectors are
the two-digit NAICS manufacturing sectors 31, 32 and 33.
25
APPENDIX B
Here we describe in detail the computational algorithm used to solve the recursive problem of the
firm. We also present robustness results discussed in the model section.
p
The state variables of the problem are last period’s real prices, i,t−1
Pt , and the current pro
p1,t−1 p2,t−1
pn,t−1
ductivity shocks, that is, p−1 =
and A = (A1,t , A2,t , ..., An,t ) . The value
Pt ,
Pt , ...,
Pt
functions are given by:
a
Z Z
V (A) = max π (p; A) − K + β
0
V p−1 , A
p
n
Z Z
V (p−1 , A) = π (p−1 ; A) + β
0
dF
1A , 2A , ..., nA
dF (P )
0
0
V p−1 , A dF 1A , 2A , ..., nA dF (P )
(B-1)
(B-2)
where V a (A) is the firm’s value of adjusting all prices, V n (p−1 , A) is the firm’s value of not
adjusting prices, 0 denotes the subsequent period, and
V = max (V a , V n ) .
Our numerical strategy to solve for the value functions consists of two major steps. First, as
described in Miranda and Fackler (2002), we approximate the value functions by projecting them
onto a polynomial space. Second, we compute the coefficients of the polynomials that are a solution
to the non-linear system of equations given by the value functions.
In particular, we approximate each value function, V a (A) and V n (p−1 , A)), by a set of higher
order Chebychev polynomials and require (B-1) and (B-2) to hold exactly at a set of points given
by the tensor product of a fixed set of collocation nodes of the state variables. This implies the
following system of non-linear equations, the so-called collocation equations:
Φa ca = v a (ca )
na na
Φ c
na
na
= v (c )
(B-3)
(B-4)
where ca and cna are basis function coefficients in the adjustment and non-adjustment cases and Φa
and Φna are the collocation matrices. These matrices are given by the value of the basis functions
evaluated at the set of nodes. The right-hand side contains the collocation functions evaluated at
the set of the collocation nodes. Note that this is the same as the value of the right-hand side of
the value functions evaluated at the collocation nodes, but where the value functions are replaced
by their approximations.
We use the same number of collocation nodes as the order of the polynomial approximation.
Therefore, we choose between 7-11 nodes for the productivity state variable and 15-20 nodes for the
real prices. Moreover, we pick the approximation range to be ± 2.5 times the standard deviation
from the mean of the underlying processes. We use Gaussian quadrature to calculate the expectations on the right-hand side, with 11-15 points for the real price transitions due to inflation while
calculating the expectations due to productivity shocks exactly. For the adjustment case, we use a
Nelder-Mead simplex method to find the maximum with an accuracy of the maximizer of 10−10 .
Next, we solve for the unknown basis function coefficients ca and cna . We express the collocation
1
equations as two fixed-point problems:
ca = Φa−1 v a (ca )
na
c
= Φ
na−1 na
(B-5)
na
v (c )
(B-6)
and iteratively update the coefficients until the collocation equations are satisfied exactly.
Our solution method is standard in the relevant literature for example as in Midrigan (2011).
We still conduct two sensitivity analyses. First, given that zero menu costs imply flex-pricing, we
verify that the approximate solution is “good” given the known analytical solution. Figure B.1
shows that optimal price policies and the price policies obtained by the approximation line up a
45-degree line in this case where we know the exact solution. The norm of the error is of order
10−9 and errors are equi-oscillatory, as is a usual property of approximations based on Chebychev
polynomials. Second, we conduct standard stochastic simulations and find that the errors between
the left- and right-hand sides of (B-1) and (B-2) at points other than the collocation nodes are on
average of the order of 10−5 or less.
Figure B.1: Analytical and Numerical Optimal Flex Price
We compute the numerical solution for optimal adjustment prices given productivity shocks and
zero menu costs. We compare to the analytical solution known in the flex price case. Errors are of
the order of 10−9 .
2
Table B.1: Results of Simulation: No Correlation, Economies of Scope
Frequency of price changes
Size of absolute price changes
Size of positive price changes
Size of negative price changes
Fraction of positive price changes
Fraction of small price changes
Kurtosis
1st Percentile
99th Percentile
1 Good
13.89%
5.42%
5.53%
-5.26%
62.92%
2.41%
1.45
-7.24%
7.45%
2 Goods
19.16%
4.37%
4.53%
-4.12%
60.46%
20.62%
1.75
-7.84%
8.16%
3 Goods
20.22%
4.02%
4.26%
-3.66%
59.33%
23.77%
1.99
-8.34%
8.96%
Synchronization measures:
Fraction, Upwards Adjustments
Fraction, Downwards Adjustments
Correlation coefficient
Menu cost
0.40%
30.22
29.39
0
0.70%
38.05
37.04
0
0.80%
We perform stochastic simulation of our model in the 1-good, 2-good and 3-good
cases and record price adjustment decisions in each case. Then, we calculate
statistics for each case as described in the text. In the 2-good and the 3-good
cases, we report the mean of the good-specific statistics. We obtain the synchronization measure from a multinomial logit regression analogous to the empirical
multinomial logit regression. We control for inflation. Menu costs are given as a
percentage of steady state revenues.
3
Table B.2: Results of Simulation: Correlation, No Economies of Scope
Frequency of price changes
Size of absolute price changes
Size of positive price changes
Size of negative price changes
Fraction of positive price changes
Fraction of small price changes
Kurtosis
1st Percentile
99th Percentile
1 Good
13.89%
5.42%
5.53%
-5.26%
62.92%
2.41%
1.45
-7.24%
7.45%
2 Goods
17.70%
4.54%
4.71%
-4.30%
61.11%
21.38%
1.77
-8.11%
8.64%
3 Goods
13.93%
4.87%
5.21%
-4.35%
61.36%
24.11%
2.03
-9.79%
10.37%
Synchronization measures:
Fraction, Upwards Adjustments
Fraction, Downwards Adjustments
Correlation coefficient
Menu cost
0.40%
31.44
30.83
0.65
0.80%
39.16
38.16
0.65
1.20%
We perform stochastic simulation of our model in the 1-good, 2-good and 3-good
cases and record price adjustment decisions in each case. Then, we calculate
statistics for each case as described in the text. In the 2-good and the 3-good
cases, we report the mean of the good-specific statistics. We obtain the synchronization measure from a multinomial logit regression analogous to the empirical
multinomial logit regression. We control for inflation. Menu costs are given as a
percentage of steady state revenues.
4
Table B.3: Results of Simulation: No Correlation, No Economies of Scope
Frequency of price changes
Size of absolute price changes
Size of positive price changes
Size of negative price changes
Fraction of positive price changes
Fraction of small price changes
Kurtosis
1st Percentile
99th Percentile
1 Good
13.89%
5.42%
5.53%
-5.26%
62.92%
2.41%
1.45
-7.24%
7.45%
2 Goods
16.25%
4.49%
4.66%
-4.23%
61.86%
21.13%
1.81
-8.01%
8.54%
3 Goods
13.93%
4.87%
5.21%
-4.35%
61.36%
24.11%
2.03
-9.79%
10.37%
Synchronization measures:
Fraction, Upwards Adjustments
Fraction, Downwards Adjustments
Correlation coefficient
Menu cost
0.40%
30.33
29.43
0
0.80%
38.40
37.11
0
1.20%
We perform stochastic simulation of our model in the 1-good, 2-good and 3-good
cases and record price adjustment decisions in each case. Then, we calculate
statistics for each case as described in the text. In the 2-good and the 3-good
cases, we report the mean of the good-specific statistics. We obtain the synchronization measure from a multinomial logit regression analogous to the empirical
multinomial logit regression. We control for inflation. Menu costs are given as a
percentage of steady state revenues.
5
Table B.4: Multiproduct vs. Multiple Single-Product Firms
Frequency of price changes
Size of absolute price changes
Size of positive price changes
Size of negative price changes
Fraction of positive price changes
Fraction of small price changes
Kurtosis
1st Percentile
99th Percentile
2 Goods
MP Firm
17.85%
4.60%
4.66%
-4.52%
62.03%
15.92%
1.72
-7.94%
8.36%
2 Firms
13.89%
5.42%
5.53%
-5.26%
62.92%
3.46%
1.45
-7.42%
7.56%
3 Goods
MP Firm
20.21%
4.18%
4.34%
-3.95%
60.31%
19.54%
2.03
-8.68%
9.04%
3 Firms
13.89%
5.42%
5.52%
-5.26%
62.92%
3.46%
1.45
-7.41%
7.56%
Synchronization measures:
Fraction, Upwards Adjustments
Fraction, Downwards Adjustments
Correlation coefficient
Menu cost
31.29
30.72
0.65
0.70%
29.52
27.46
0
0.40%
38.65
37.72
0.65
0.80%
14.54
14.05
0
0.40%
We perform stochastic simulation of our model for the 2-good and 3-good multi-product firms as
in Table 4. Results from these simulations are summarized under the columns “MP Firms.” In
addition, we simulate two, and respectively three 1-good firms subject to common inflationary
shocks but completely independent productivity draws. We record price adjustment decisions
and calculate statistics for each case as described in the text. In the 2-good and the 3-good cases,
we report the mean of the good-specific statistics. We obtain the synchronization measure from a
multinomial logit regression analogous to the empirical multinomial logit regression. We control
for inflation. Menu costs are given as a percentage of steady state revenues.
6
APPENDIX C
In this appendix, we describe in further detail the sampling procedure of the BLS which implies
a monotonic relationship between the actual and sampled number of goods produced by multiproduct firms. First, we document that firms with more goods have larger total sales. More goods
will therefore be sampled in total from large firms. Second, we show that sales shift to goods with
lower sales rank in firms with more goods. Therefore, standard survey design implies that not only
more, but different goods will be sampled from larger firms. Finally, we also summarize the fraction
of joint price changes in firms.
Sales Values
First, we document that firms with more goods have larger total sales in the data. Because total
sales value determines the sampling probabilities of firms in the sampling selection procedure,11
this implies that on average more goods will therefore be sampled from large firms.
We compute our measure of total sales value for each n-good-type firm in the following way.
First, we compute the total dollar-value sales in a given month, year, and firm by aggregating up
the item dollar-value of sales from the last time the item was re-sampled. Second, we count the
number of goods for each firm in a given month and year. Third, we compute the unweighted and
weighted median total sales value across all firms for a given n-good type of firm and month and
year. Fourth, we calculate the mean and median sales value for an n-good type of firm
We find that firms with more goods have larger total sales: there is a strong empirical, monotonic
relationship between the number of goods and the natural logarithm of the total sales. Figure C.1
summarizes this relationship. Because total sales value determines the sampling probabilities of
firms in the BLS sampling selection procedure, on average a higher number of goods will be collected
from large firms.
Within-Firm Sales Shares
Second, we show that sales shift to goods with lower sales rank in firms with more goods.
Therefore, standard survey design such as sampling proportional to size implies that not only
more, but different goods will likely be sampled from larger firms.
We compute within-firm sales shares and sales ranks in the following way. First, we compute the
total dollar-value sales for a given month, year, and firm by aggregating up the dollar-value of sales
of the good from the last time the item was re-sampled. Second, we calculate the good-specific sales
shares for each firm in a given month and year. Third, we rank the goods in each firm according to
these sales shares. Fourth, we count the number of goods for each firm in a given month and year.
Fifth, we compute the mean sales shares for an r-ranked good in an n-good firm in a given month
and year, across all firms. Sixth, we compute the sales-weighted mean for an r-ranked good in
an n-good firm over time. These calculations give us the sales share representative of an r-ranked
good in a firm with n goods.12
We find that sales shift to goods with lower sales rank in firms with more goods. For example,
the representative sales share of the best-selling good in a two-good firm is 63% while it is 45%
for the second good. For a three-good firm, the sales shares are 45%, 35%, and 30%. Table C.1
11
Employment is another measure of firm size. The exact same results hold for employment: firms with more goods
have a larger number of employees.
12
Note that these shares do not have to sum up to 100% in an n-good firm by way of computation.
1
summarizes sales shares by the number of goods and rank of the goods. The table covers firms
with up to 11 goods which account for more than 98% of all prices in the data. Under standard
survey designs such as sampling proportional to size and a fixed survey budget not only more, but
different goods are more likely to be sampled when firms produce more goods.
Log Mean Firm Sales Value by Number of Goods
21
20.5
Log Mean Sales Value
20
19.5
19
18.5
18
17.5
17
16.5
16
1
2
3
4
5
6
7
8
9
Number of Goods
Figure C.1: Log Mean Firm Sales Value by Number of Goods
2
10
11
3
2
62.861%
(0.131)%
44.952%
(0.081%)
1
100.000%
(0.000)%
45.110%
(0.075)%
34.453%
(0.051%)
29.499%
(0.028%)
3
35.916%
(0.086)%
27.594%
(0.034%)
24.051%
(0.030%)
22.360%
(0.030%)
4
37.090%
(0.413)%
22.966%
(0.035%)
19.703%
(0.023%)
18.251%
(0.039%)
17.579%
(0.044%)
5
28.301%
(0.097)%
19.963%
(0.049%)
17.428%
(0.031%)
16.162%
(0.013%)
15.197%
(0.007%)
14.748%
(0.011%)
6
25.517%
(0.337)%
18.396%
(0.170%)
15.896%
(0.032%)
14.357%
(0.034%)
13.577%
(0.028%)
12.629%
(0.038%)
12.356%
(1.261%)
7
19.084%
(0.381)%
15.209%
(0.091%)
14.223%
(0.069%)
13.318%
(0.050%)
12.707%
(0.052%)
11.771%
(0.016%)
11.417%
(1.165%)
11.302%
(1.153%)
8
25.532%
(0.480)%
15.749%
(0.260%)
13.160%
(0.134%)
11.375%
(0.049%)
10.600%
(0.066%)
10.185%
(0.070%)
10.072%
(1.028%)
9.679%
(0.988%)
9.943%
(1.015%)
9
14.393%
(0.344)%
12.297%
(0.089%)
11.316%
(0.073%)
10.615%
(0.038%)
10.212%
(0.028%)
9.824%
(0.032%)
9.578%
(0.978%)
9.357%
(0.955%)
9.257%
(0.945%)
9.222%
(0.027%)
10
16.642%
(0.427)
10.748%
(0.106%)
10.514%
(0.105%)
9.835%
(0.057%)
9.393%
(0.032%)
9.051%
(0.017%)
8.924%
(0.911)
8.731%
(0.891%)
8.562
(0.874%)
8.456%
(0.064%)
8.528%
(0.057%)
11
and year, across all rms. Sixth, we compute the sales-weighted mean for an r-ranked good in an n-good rm over time.
we count the number of goods for each rm in a given month and year. Fifth, we compute mean sales shares for an r-ranked good in an n-good rm in a given month
Second, we calculate the good-specic sales shares for each rm in a given month and year. Third, we rank goods in each rm according to the sales shares. Fourth,
Based on the PPI data, we first compute the total dollar-value sales for a given month, year, and rm by aggregating up the dollar-value of sales of each good.
11
10
9
8
7
6
5
4
3
2
#Goods
/#Rank
1
Table C.1: Mean Sales Shares of r-Ranked Goods for n-Good Firms, Sales-Weighted